NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 10.0s

Alternatives: 6

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: 78.4% 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: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{b + a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ 0.5 (* b (/ a PI))) (+ b a)))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(0.5 / Float64(b * Float64(a / pi))) / Float64(b + a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 / (b * (a / pi))) / (b + a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(0.5 / N[(b * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 2: 71.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(b \cdot \left(-a\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 9.2e+93)
   (/ (* 0.5 (/ PI a)) (* b a))
   (/ (* 0.5 PI) (* b (* b (- a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+93) {
		tmp = (0.5 * (((double) M_PI) / a)) / (b * a);
	} else {
		tmp = (0.5 * ((double) M_PI)) / (b * (b * -a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+93) {
		tmp = (0.5 * (Math.PI / a)) / (b * a);
	} else {
		tmp = (0.5 * Math.PI) / (b * (b * -a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 9.2e+93:
		tmp = (0.5 * (math.pi / a)) / (b * a)
	else:
		tmp = (0.5 * math.pi) / (b * (b * -a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 9.2e+93)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(b * a));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(b * Float64(-a))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.2e+93)
		tmp = (0.5 * (pi / a)) / (b * a);
	else
		tmp = (0.5 * pi) / (b * (b * -a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 9.2e+93], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 3: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e-235)
   (* (/ PI a) (/ 0.5 (* b a)))
   (* (/ PI (* b a)) (/ -0.5 a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-235) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	} else {
		tmp = (((double) M_PI) / (b * a)) * (-0.5 / a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-235) {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	} else {
		tmp = (Math.PI / (b * a)) * (-0.5 / a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -8.5e-235:
		tmp = (math.pi / a) * (0.5 / (b * a))
	else:
		tmp = (math.pi / (b * a)) * (-0.5 / a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -8.5e-235)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(-0.5 / a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e-235)
		tmp = (pi / a) * (0.5 / (b * a));
	else
		tmp = (pi / (b * a)) * (-0.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -8.5e-235], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 4: 60.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-236}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.6e-236)
   (/ (* 0.5 (/ PI a)) (* b a))
   (* (/ PI (* b a)) (/ -0.5 a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.6e-236) {
		tmp = (0.5 * (((double) M_PI) / a)) / (b * a);
	} else {
		tmp = (((double) M_PI) / (b * a)) * (-0.5 / a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.6e-236) {
		tmp = (0.5 * (Math.PI / a)) / (b * a);
	} else {
		tmp = (Math.PI / (b * a)) * (-0.5 / a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.6e-236:
		tmp = (0.5 * (math.pi / a)) / (b * a)
	else:
		tmp = (math.pi / (b * a)) * (-0.5 / a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.6e-236)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(b * a));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(-0.5 / a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.6e-236)
		tmp = (0.5 * (pi / a)) / (b * a);
	else
		tmp = (pi / (b * a)) * (-0.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -6.6e-236], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{0.5}{b + a} \cdot \frac{\frac{\pi}{a}}{b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ 0.5 (+ b a)) (/ (/ PI a) b)))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(0.5 / Float64(b + a)) * Float64(Float64(pi / a) / b))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 / (b + a)) * ((pi / a) / b);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 6: 63.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\pi}{a} \cdot \frac{0.5}{b \cdot a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ PI a) (/ 0.5 (* b a))))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (math.pi / a) * (0.5 / (b * a))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (pi / a) * (0.5 / (b * a));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

Reproduce

herbie shell --seed 2024008 
(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))))