math.h

05/main.c

@@ -1,15 +1,9 @@
 #define _STDLIB_DEBUG
+#include <math.h>
 #include <stdio.h>
-#include <string.h>
-#include <stdlib.h>
-#include <stddef.h>
-#include <ctype.h>
-#include <locale.h>
 
 int main(int argc, char **argv) {
-	setlocale(LC_ALL, "C");
+	printf("%.16g\n", cosh(10));
-	struct lconv *c = localeconv();
-	printf("%s\n",c->negative_sign);
 	return 0;
 }
 

05/math.h

@@ -0,0 +1,409 @@
+#ifndef _MATH_H
+#define _MATH_H
+
+#include <stdc_common.h>
+#define HUGE_VAL _INFINITY // glibc defines HUGE_VAL as infinity (the C standard only requires it to be positive, funnily enough)
+#define _NAN (-(_INFINITY-_INFINITY))
+#define _PI 3.141592653589793
+#define _2PI 6.283185307179586
+#define _HALF_PI 1.5707963267948966
+#define _THREE_HALVES_PI 4.71238898038469
+
+// NOTE: these functions are not IEEE 754-compliant (the C standard doesn't require them to be), but they're pretty good
+
+double frexp(double value, int *exp) {
+	if (value == 0) {
+		*exp = 0;
+		return 0;
+	}
+	unsigned long u = *(unsigned long *)&value, significand;
+	*exp = ((u >> 52) & 0x7ff) - 1022;
+	// replace exponent with 1022
+	u &= 0x800fffffffffffff;
+	u |= 0x3fe0000000000000;
+	return *(double *)&u;
+}
+
+double ldexp(double x, int exp) {
+	int e;
+	double y = frexp(x, &e);
+	// since x = y * 2^e,  x * 2^exp = y * 2^(e+exp)
+	exp += e;
+	if (exp < -1022) return 0;
+	if (exp > 1023) return _INFINITY;
+	unsigned long pow2 = (unsigned long)(exp + 1023) << 52;
+	return y * *(double *)&pow2;
+}
+
+double floor(double x) {
+	if (x >= 0.0) {
+		if (x > 1073741824.0 * 1073741824.0)
+			return x; // floats this big must be integers
+		return (unsigned long)x;
+	} else {
+		if (x < -1073741824.0 * 1073741824.0)
+			return x; // floats this big must be integers
+		double i = (long)x;
+		if (x == i) return x;
+		return i - 1.0;
+	}
+}
+
+double ceil(double x) {
+	double f = floor(x);
+	if (x == f) return f;
+	return f + 1.;
+}
+
+double fabs(double x) {
+	// this is better than x >= 0 ? x : -x because it deals with -0 properly
+	unsigned long u = *(unsigned long *)&x;
+	u &= 0x7fffffffffffffff;
+	return *(double *)&u;
+}
+
+double fmod(double x, double y) {
+	if (y == 0.0) {
+		errno = EDOM;
+		return 0.0;
+	}
+	return x - (floor(x / y) * y);
+}
+
+double _sin_taylor(double x) {
+	double i;
+	double term = x;
+	// taylor expansion for sin:   x - x³/3! + x⁵/5! - ...
+	
+	// https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+	double prev = -1.0;
+	double sum = 0.0;
+	double c = 0.0;
+	for (i = 0.0; i < 100.0 && sum != prev; ++i) {
+		prev = sum;
+		double y = term - c;
+		double t = sum + y;
+		c = (t - sum) - y;
+		sum = t;
+		term *= -(x * x) / ((2.0*i+2.0)*(2.0*i+3.0));
+	}
+	return sum;
+}
+
+double _cos_taylor(double x) {
+	double i;
+	double term = 1.0;
+	// taylor expansion for cos:   1 - x²/2! + x⁴/4! - ...
+	
+	// https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+	double prev = -1.0;
+	double sum = 0.0;
+	double c = 0.0;
+	for (i = 0.0; i < 100.0 && sum != prev; ++i) {
+		prev = sum;
+		double y = term - c;
+		double t = sum + y;
+		c = (t - sum) - y;
+		sum = t;
+		term *= -(x * x) / ((2.0*i+1.0)*(2.0*i+2.0));
+	}
+	return sum;
+}
+
+double sin(double x) {
+	x = fmod(x, 2.0*_PI);
+	// the Taylor series works best for small inputs. so, provide _sin_taylor with a value in the range [0,π/2]
+	if (x < _HALF_PI)
+		return _sin_taylor(x);
+	if (x < _PI)
+		return _sin_taylor(_PI - x);
+	if (x < _THREE_HALVES_PI)
+		return -_sin_taylor(x - _PI);
+	return -_sin_taylor(_2PI - x);
+}
+
+double cos(double x) {
+	x = fmod(x, 2.0*_PI);
+	// the Taylor series works best for small inputs. so, provide _cos_taylor with a value in the range [0,π/2]
+	if (x < _HALF_PI)
+		return _cos_taylor(x);
+	if (x < _PI)
+		return -_cos_taylor(_PI - x);
+	if (x < _THREE_HALVES_PI)
+		return -_cos_taylor(x - _PI);
+	return _cos_taylor(_2PI - x);
+}
+
+double tan(double x) {
+	return sin(x)/cos(x);
+}
+
+// for sqrt and the inverse trigonometric functions, we use Newton's method
+// https://en.wikipedia.org/wiki/Newton%27s_method
+
+double sqrt(double x) {
+	if (x < 0.0) {
+		errno = EDOM;
+		return _NAN;
+	}
+	if (x == 0.0) return 0.0;
+	if (x == _INFINITY) return _INFINITY;
+	// we want to find the root of: f(t) = t² - x
+	//                              f'(t) = 2t
+	int exp;
+	double y = frexp(x, &exp);
+	if (exp & 1) {
+		y *= 2;
+		--exp;
+	}
+	// newton's method will be slow for very small or very large numbers.
+	// so we have ensured that
+	//     0.5 ≤ y < 2
+	// and also x = y * 2^exp; sqrt(x) = sqrt(y) * 2^(exp/2)  NB: we've ensured that exp is even
+	
+	// 7 iterations seems to be more than enough for any number
+	double t = y;
+	t = (y / t + t) * 0.5;
+	t = (y / t + t) * 0.5;
+	t = (y / t + t) * 0.5;
+	t = (y / t + t) * 0.5;
+	t = (y / t + t) * 0.5;
+	t = (y / t + t) * 0.5;
+	t = (y / t + t) * 0.5;
+	
+	return ldexp(t, exp>>1);
+	
+}
+
+double _acos_newton(double x) {
+	// we want to find the root of: f(t) = cos(t) - x
+	//                              f'(t) = -sin(t)
+	double t = _HALF_PI - x; // reasonably good first approximation
+	double prev_t = -100.0;
+	int i;
+	
+	for (i = 0; i < 100 && prev_t != t; ++i) {
+		prev_t = t;
+		t += (cos(t) - x) / sin(t);
+	}
+	return t;
+}
+
+double _asin_newton(double x) {
+	// we want to find the root of: f(t) = sin(t) - x
+	//                              f'(t) = cos(t)
+	double t = x; // reasonably good first approximation
+	double prev_t = -100.0;
+	int i;
+	
+	for (i = 0; i < 100 && prev_t != t; ++i) {
+		prev_t = t;
+		t += (x - sin(t)) / cos(t);
+	}
+	return t;
+}
+
+double acos(double x) {
+	if (x > 1.0 || x < -1.0) {
+		errno = EDOM;
+		return _NAN;
+	}
+	// Newton's method doesn't work well near -1 and 1, because f(x) / f'(x) is very large.
+	if (x > 0.8)
+		return _asin_newton(sqrt(1-x*x));
+	if (x < -0.8)
+		return _PI-_asin_newton(sqrt(1-x*x));
+	
+	return _acos_newton(x);
+}
+
+double asin(double x) {
+	if (x > 1.0 || x < -1.0) {
+		errno = EDOM;
+		return _NAN;
+	}
+	// Newton's method doesn't work well near -1 and 1, because f(x) / f'(x) is very large.
+	if (x > 0.8)
+		return _acos_newton(sqrt(1.0-x*x));
+	if (x < -0.8)
+		return -_acos_newton(sqrt(1.0-x*x));
+	
+	return _asin_newton(x);
+}
+
+double atan(double x) {
+	// the formula below breaks for really large inputs; tan(10^20) as a double is indistinguishable from pi/2 anyways
+	if (x > 1e20) return _HALF_PI;
+	if (x < -1e20) return -_HALF_PI;
+	
+	// we can use a nice trigonometric identity here
+	return asin(x / sqrt(1+x*x));
+}
+
+double atan2(double y, double x) {
+	if (x == 0.0) {
+		if (y > 0.0) return _HALF_PI;
+		if (y < 0.0) return -_HALF_PI;
+		return 0.0; // this is what IEEE 754 does
+	}
+	
+	double a = atan(y/x);
+	if (x > 0.0) {
+		return a;
+	} else if (y > 0.0) {
+		return a + _PI;
+	} else {
+		return a - _PI;
+	}
+}
+
+double _exp_taylor(double x) {
+	double i;
+	double term = 1.0;
+	// taylor expansion for exp:   1 + x/1! + x²/2! + ...
+	
+	// https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+	double prev = -1.0;
+	double sum = 0.0;
+	double c = 0.0;
+	for (i = 1.0; i < 100.0 && sum != prev; ++i) {
+		prev = sum;
+		double y = term - c;
+		double t = sum + y;
+		c = (t - sum) - y;
+		sum = t;
+		term *= x / i;
+	}
+	return sum;
+}
+
+double exp(double x) {
+	if (x > 709.782712893384) {
+		errno = ERANGE;
+		return _INFINITY;
+	}
+	if (x == 0.0) return 1;
+	if (x < -744.4400719213812)
+		return 0;
+	int i, e;
+	double y = frexp(x, &e);
+	if (e < 1.0) return _exp_taylor(x);
+	// the taylor series doesn't work well for large x (positive or negative),
+	// so we use the fact that   exp(y*2^e) = exp(y)^(2^e)
+	double value = _exp_taylor(y);
+	for (i = 0; i < e; ++i)
+		value *= value;
+	return value;
+}
+
+#define _LOG2 0.6931471805599453
+
+double log(double x) {
+	if (x < 0.0) {
+		errno = EDOM;
+		return _NAN;
+	}
+	if (x == 0.0) return -_INFINITY;
+	if (x == 1.0) return 0.0;
+	int e;
+	double sum;
+	double a = frexp(x, &e);
+	// since x = a * 2^e, log(x) = log(a) + log(2^e) = log(a) + e log(2)
+	sum = e * _LOG2;
+	// now that a is in [1/2,1), the series log(a) = (a-1) - (a-1)²/2 + (a-1)³/3 - ... converges quickly
+	
+	a -= 1;
+	// https://en.wikipedia.org/wiki/Kahan_summation_algorithm
+	double prev = HUGE_VAL;
+	double c = 0;
+	double term = a;
+	double i;
+	for (i = 1.0; i < 100.0 && sum != prev; ++i) {
+		prev = sum;
+		double y = term / i - c;
+		double t = sum + y;
+		c = (t - sum) - y;
+		sum = t;
+		term *= -a;
+	}
+	return sum;
+}
+
+#define _INVLOG10 0.43429448190325176 // = 1/log(10)
+double log10(double x) {
+	return log(x) * _INVLOG10;
+}
+
+double modf(double value, double *iptr) {
+	double m = fmod(value, 1.0);
+	if (value >= 0.0) {
+		*iptr = value - m;
+		return m;
+	} else if (m == 0.0) {
+		*iptr = value;
+		return 0.0;
+	} else {
+		*iptr = value - m + 1.0;
+		return m - 1.0;
+	}
+}
+
+// double raised to the power of an integer
+double _dpowi(double x, unsigned long y) {
+	double result = 1.0;
+	if (y & 1) {
+		--y;
+		result *= x;
+	}
+	if (y > 0) {
+		double p = _dpowi(x, y >> 1);
+		result *= p * p;
+	}
+	return result;
+}
+
+double pow(double x, double y) {
+	if (x > 0.0) {
+		return exp(y * log(x));
+	} else if (x < 0.0) {
+		if (fmod(y, 1.0) != 0) {
+			errno = EDOM;
+			return _NAN;
+		}
+		if (y > 1.6602069666338597e+19)
+			return x < -1. ? -_INFINITY : 0.;
+		if (y < -1.6602069666338597e+19)
+			return x < -1. ? 0. : -_INFINITY;
+		return _dpowi(x, (unsigned long)y);
+	} else {
+		if (y < 0) {
+			errno = EDOM;
+			return _NAN;
+		}
+		if (y > 0) {
+			// 0^x = 0 for x>0
+			return 0.;
+		}
+		// 0^0 = 1
+		return 1.;
+	}
+}
+
+double cosh(double x) {
+	double e = exp(x);
+	return (e + 1./e) * 0.5;
+}
+
+double sinh(double x) {
+	double e = exp(x);
+	return (e - 1./e) * 0.5;
+}
+
+double tanh(double x) {
+	if (x > 20.0) return 1.;
+	if (x < -20.0) return -1.;
+	double e = exp(x);
+	double f = 1./e;
+	return (e - f) / (e + f);
+}
+#endif // _MATH_H

05/stdc_common.h

@@ -424,6 +424,7 @@ double strtod(const char *nptr, char **endptr) {
 		sum = t;
 	}
 	
+	if (sum == _INFINITY) errno = ERANGE;
 	if (endptr) *endptr = nptr;
 	return sum * sign;
 }

05/tokenize.b

@@ -287,6 +287,7 @@ function tokenize
 			fraction = strtoi(&in, 10)
 			; e.g. to turn 35 into .35, multiply by 10^-2
 			pow10 = p - in
+			;putnln_signed(pow10)
 			if pow10 < -400 goto bad_float
 			:float_no_fraction
 			; construct the number integer + fraction*10^pow10
@@ -303,16 +304,13 @@ function tokenize
 			exponent = new_exponent
 			significand += right_shift(integer, exponent)
 			:float_no_integer
+			
 			if *1in == 'e goto float_exponent
 			if *1in == 'E goto float_exponent
 			
 			:float_have_significand_and_exponent
 			if significand == 0 goto float_zero
 			normalize_float(&significand, &exponent)
-			; putn(significand)
-			; putc(32)
-			; putn_signed(exponent)
-			; putc(10)
 			; make number round to the nearest representable float roughly (this is what gcc does)
 			; this fails for 5e-100 probably because of imprecision, but mostly works
 			significand += 15
@@ -321,6 +319,7 @@ function tokenize
 			exponent += 5
 			exponent += 52  ; 1001010111... => 1.001010111... 
 			n = leftmost_1bit(significand)
+			exponent += n - 52 ; in most cases, this is 0, but sometimes it turns out to be 1.
 			b = 1 < n
 			significand &= ~b
 			data = significand