NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.7%

Time: 10.1s

Alternatives: 8

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \frac{\pi}{a \cdot b}}{a + b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (* 0.5 (/ PI (* a b))) (+ a b)))

C

double code(double a, double b) {
	return (0.5 * (((double) M_PI) / (a * b))) / (a + b);
}

Java

public static double code(double a, double b) {
	return (0.5 * (Math.PI / (a * b))) / (a + b);
}

Python

def code(a, b):
	return (0.5 * (math.pi / (a * b))) / (a + b)

Julia

function code(a, b)
	return Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(a + b))
end

MATLAB

function tmp = code(a, b)
	tmp = (0.5 * (pi / (a * b))) / (a + b);
end

Wolfram

code[a_, b_] := N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 77.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.488 \cdot 10^{-61}:\\ \;\;\;\;\frac{-0.5}{a \cdot b} \cdot \frac{-\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.488e-61)
   (* (/ -0.5 (* a b)) (/ (- PI) a))
   (* (/ PI (* a b)) (/ 0.5 (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.488e-61) {
		tmp = (-0.5 / (a * b)) * (-((double) M_PI) / a);
	} else {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.488e-61) {
		tmp = (-0.5 / (a * b)) * (-Math.PI / a);
	} else {
		tmp = (Math.PI / (a * b)) * (0.5 / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.488e-61:
		tmp = (-0.5 / (a * b)) * (-math.pi / a)
	else:
		tmp = (math.pi / (a * b)) * (0.5 / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.488e-61)
		tmp = Float64(Float64(-0.5 / Float64(a * b)) * Float64(Float64(-pi) / a));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.488e-61)
		tmp = (-0.5 / (a * b)) * (-pi / a);
	else
		tmp = (pi / (a * b)) * (0.5 / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.488e-61], N[(N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[((-Pi) / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{-0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e-83)
   (* (/ PI (- b a)) (/ -0.5 (* a b)))
   (/ (* 0.5 (/ PI (* a b))) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-83) {
		tmp = (((double) M_PI) / (b - a)) * (-0.5 / (a * b));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-83) {
		tmp = (Math.PI / (b - a)) * (-0.5 / (a * b));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.5e-83:
		tmp = (math.pi / (b - a)) * (-0.5 / (a * b))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e-83)
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(-0.5 / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e-83)
		tmp = (pi / (b - a)) * (-0.5 / (a * b));
	else
		tmp = (0.5 * (pi / (a * b))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e-83], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -41000000000000:\\ \;\;\;\;\frac{-0.5}{a \cdot b} \cdot \frac{-\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -41000000000000.0)
   (* (/ -0.5 (* a b)) (/ (- PI) a))
   (/ (* 0.5 (/ PI (* a b))) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -41000000000000.0) {
		tmp = (-0.5 / (a * b)) * (-((double) M_PI) / a);
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -41000000000000.0) {
		tmp = (-0.5 / (a * b)) * (-Math.PI / a);
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -41000000000000.0:
		tmp = (-0.5 / (a * b)) * (-math.pi / a)
	else:
		tmp = (0.5 * (math.pi / (a * b))) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -41000000000000.0)
		tmp = Float64(Float64(-0.5 / Float64(a * b)) * Float64(Float64(-pi) / a));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -41000000000000.0)
		tmp = (-0.5 / (a * b)) * (-pi / a);
	else
		tmp = (0.5 * (pi / (a * b))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -41000000000000.0], N[(N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[((-Pi) / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a + b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ (/ PI b) a) (/ 0.5 (+ a b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / b) / a) * (0.5 / (a + b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / b) / a) * (0.5 / (a + b));
}

Python

def code(a, b):
	return ((math.pi / b) / a) * (0.5 / (a + b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / b) / a) * Float64(0.5 / Float64(a + b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / b) / a) * (0.5 / (a + b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 30.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{b} \cdot \frac{-0.5}{a \cdot b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ PI b) (/ -0.5 (* a b))))

C

double code(double a, double b) {
	return (((double) M_PI) / b) * (-0.5 / (a * b));
}

Java

public static double code(double a, double b) {
	return (Math.PI / b) * (-0.5 / (a * b));
}

Python

def code(a, b):
	return (math.pi / b) * (-0.5 / (a * b))

Julia

function code(a, b)
	return Float64(Float64(pi / b) * Float64(-0.5 / Float64(a * b)))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi / b) * (-0.5 / (a * b));
end

Wolfram

code[a_, b_] := N[(N[(Pi / b), $MachinePrecision] * N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 7: 30.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\left(a \cdot b\right) \cdot \frac{b}{\pi}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ -0.5 (* (* a b) (/ b PI))))

C

double code(double a, double b) {
	return -0.5 / ((a * b) * (b / ((double) M_PI)));
}

Java

public static double code(double a, double b) {
	return -0.5 / ((a * b) * (b / Math.PI));
}

Python

def code(a, b):
	return -0.5 / ((a * b) * (b / math.pi))

Julia

function code(a, b)
	return Float64(-0.5 / Float64(Float64(a * b) * Float64(b / pi)))
end

MATLAB

function tmp = code(a, b)
	tmp = -0.5 / ((a * b) * (b / pi));
end

Wolfram

code[a_, b_] := N[(-0.5 / N[(N[(a * b), $MachinePrecision] * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (* 0.5 (/ PI (* a b))) b))

C

double code(double a, double b) {
	return (0.5 * (((double) M_PI) / (a * b))) / b;
}

Java

public static double code(double a, double b) {
	return (0.5 * (Math.PI / (a * b))) / b;
}

Python

def code(a, b):
	return (0.5 * (math.pi / (a * b))) / b

Julia

function code(a, b)
	return Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / b)
end

MATLAB

function tmp = code(a, b)
	tmp = (0.5 * (pi / (a * b))) / b;
end

Wolfram

code[a_, b_] := N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023341 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))