NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 96.7%

Time: 9.4s

Alternatives: 7

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: 78.8% 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: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{\pi}{b + a}}{-b} \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e+91)
   (* (/ (/ PI (+ b a)) (- b)) (/ -0.5 a))
   (/ (/ (* PI 0.5) b) (* b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+91) {
		tmp = ((((double) M_PI) / (b + a)) / -b) * (-0.5 / a);
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) / (b * a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+91) {
		tmp = ((Math.PI / (b + a)) / -b) * (-0.5 / a);
	} else {
		tmp = ((Math.PI * 0.5) / b) / (b * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e+91:
		tmp = ((math.pi / (b + a)) / -b) * (-0.5 / a)
	else:
		tmp = ((math.pi * 0.5) / b) / (b * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+91)
		tmp = Float64(Float64(Float64(pi / Float64(b + a)) / Float64(-b)) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) / Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+91)
		tmp = ((pi / (b + a)) / -b) * (-0.5 / a);
	else
		tmp = ((pi * 0.5) / b) / (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+91], N[(N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0000000000000002e91

   1. Initial program 82.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 82.8%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 82.8%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 82.8%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 89.9%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 90.1%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]

   4. Taylor expanded in a around inf 69.8%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{-1}{b}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 55.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b}\right)\right)} \]

      2. expm1-udef 47.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b}\right)} - 1} \]

      3. *-commutative 47.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{b} \cdot \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right)}\right)} - 1 \]

      4. frac-2neg 47.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{--1}{-b}} \cdot \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right)\right)} - 1 \]

      5. metadata-eval 47.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{-b} \cdot \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right)\right)} - 1 \]

      6. associate-*l/ 47.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{-b} \cdot \color{blue}{\frac{\frac{\pi}{b + a} \cdot 0.5}{b - a}}\right)} - 1 \]

      7. frac-times 47.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(\frac{\pi}{b + a} \cdot 0.5\right)}{\left(-b\right) \cdot \left(b - a\right)}}\right)} - 1 \]

      8. *-un-lft-identity 47.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{\pi}{b + a} \cdot 0.5}}{\left(-b\right) \cdot \left(b - a\right)}\right)} - 1 \]

   6. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{b + a} \cdot 0.5}{\left(-b\right) \cdot \left(b - a\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 63.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{b + a} \cdot 0.5}{\left(-b\right) \cdot \left(b - a\right)}\right)\right)} \]

      2. expm1-log1p 77.4%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a} \cdot 0.5}{\left(-b\right) \cdot \left(b - a\right)}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a}}{-b} \cdot \frac{0.5}{b - a}} \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a}}{-b} \cdot \frac{0.5}{b - a}} \]

   9. Taylor expanded in b around 0 97.7%

      \[\leadsto \frac{\frac{\pi}{b + a}}{-b} \cdot \color{blue}{\frac{-0.5}{a}} \]

   if 5.0000000000000002e91 < b

   1. Initial program 70.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 70.8%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 70.8%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 70.8%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 84.8%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 84.9%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 84.9%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 84.9%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 84.9%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 84.9%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 84.9%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. inv-pow 84.9%

         \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   5. Applied egg-rr 84.9%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 84.9%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. +-commutative 84.9%

         \[\leadsto \left(\frac{\frac{1}{\frac{\color{blue}{a + b}}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   7. Simplified 84.9%

      \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{a + b}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   8. Taylor expanded in a around 0 84.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 84.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 84.9%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 84.9%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 84.9%

         \[\leadsto \frac{\pi}{a} \cdot \frac{0.5}{\color{blue}{b \cdot b}} \]

   10. Simplified 84.9%

       \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]
   11. Step-by-step derivation

       1. frac-times 84.9%

          \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}} \]

       2. *-commutative 84.9%

          \[\leadsto \frac{\color{blue}{0.5 \cdot \pi}}{a \cdot \left(b \cdot b\right)} \]

       3. *-un-lft-identity 84.9%

          \[\leadsto \frac{\color{blue}{1 \cdot \left(0.5 \cdot \pi\right)}}{a \cdot \left(b \cdot b\right)} \]

       4. frac-times 84.8%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{0.5 \cdot \pi}{b \cdot b}} \]

       5. associate-/r* 84.9%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{\frac{0.5 \cdot \pi}{b}}{b}} \]

       6. frac-times 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{0.5 \cdot \pi}{b}}{a \cdot b}} \]

       7. *-un-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{b}}}{a \cdot b} \]

       8. *-commutative 99.8%

          \[\leadsto \frac{\frac{\color{blue}{\pi \cdot 0.5}}{b}}{a \cdot b} \]

       9. *-commutative 99.8%

          \[\leadsto \frac{\frac{\pi \cdot 0.5}{b}}{\color{blue}{b \cdot a}} \]

   12. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b}}{b \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{\pi}{b + a}}{-b} \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{b \cdot a}\\ \end{array} \]

Alternative 2: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-72} \lor \neg \left(a \leq 1.06 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{-\pi}{b \cdot a} \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{b \cdot \frac{a}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.25e-72) (not (<= a 1.06e-46)))
   (* (/ (- PI) (* b a)) (/ 0.5 (- b a)))
   (/ (/ 0.5 b) (* b (/ a PI)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.25e-72) || !(a <= 1.06e-46)) {
		tmp = (-((double) M_PI) / (b * a)) * (0.5 / (b - a));
	} else {
		tmp = (0.5 / b) / (b * (a / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.25e-72) || !(a <= 1.06e-46)) {
		tmp = (-Math.PI / (b * a)) * (0.5 / (b - a));
	} else {
		tmp = (0.5 / b) / (b * (a / Math.PI));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.25e-72) or not (a <= 1.06e-46):
		tmp = (-math.pi / (b * a)) * (0.5 / (b - a))
	else:
		tmp = (0.5 / b) / (b * (a / math.pi))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.25e-72) || !(a <= 1.06e-46))
		tmp = Float64(Float64(Float64(-pi) / Float64(b * a)) * Float64(0.5 / Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(b * Float64(a / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.25e-72) || ~((a <= 1.06e-46)))
		tmp = (-pi / (b * a)) * (0.5 / (b - a));
	else
		tmp = (0.5 / b) / (b * (a / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.25e-72], N[Not[LessEqual[a, 1.06e-46]], $MachinePrecision]], N[(N[((-Pi) / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(b * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.25e-72 or 1.06e-46 < a

   1. Initial program 80.4%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 80.4%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 80.4%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 80.4%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 89.8%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 90.1%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 90.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]

   4. Taylor expanded in a around inf 83.7%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{-1}{b}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 73.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b}\right)\right)} \]

      2. expm1-udef 63.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b}\right)} - 1} \]

      3. *-commutative 63.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{b} \cdot \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right)}\right)} - 1 \]

      4. frac-2neg 63.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{--1}{-b}} \cdot \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right)\right)} - 1 \]

      5. metadata-eval 63.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{-b} \cdot \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right)\right)} - 1 \]

      6. associate-*l/ 63.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{-b} \cdot \color{blue}{\frac{\frac{\pi}{b + a} \cdot 0.5}{b - a}}\right)} - 1 \]

      7. frac-times 63.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(\frac{\pi}{b + a} \cdot 0.5\right)}{\left(-b\right) \cdot \left(b - a\right)}}\right)} - 1 \]

      8. *-un-lft-identity 63.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{\pi}{b + a} \cdot 0.5}}{\left(-b\right) \cdot \left(b - a\right)}\right)} - 1 \]

   6. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{b + a} \cdot 0.5}{\left(-b\right) \cdot \left(b - a\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 82.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{b + a} \cdot 0.5}{\left(-b\right) \cdot \left(b - a\right)}\right)\right)} \]

      2. expm1-log1p 93.3%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a} \cdot 0.5}{\left(-b\right) \cdot \left(b - a\right)}} \]

      3. times-frac 93.2%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a}}{-b} \cdot \frac{0.5}{b - a}} \]

   8. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{b + a}}{-b} \cdot \frac{0.5}{b - a}} \]

   9. Taylor expanded in b around 0 93.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\pi}{a \cdot b}\right)} \cdot \frac{0.5}{b - a} \]
   10. Step-by-step derivation

       1. associate-*r/ 93.0%

          \[\leadsto \color{blue}{\frac{-1 \cdot \pi}{a \cdot b}} \cdot \frac{0.5}{b - a} \]

       2. mul-1-neg 93.0%

          \[\leadsto \frac{\color{blue}{-\pi}}{a \cdot b} \cdot \frac{0.5}{b - a} \]

   11. Simplified 93.0%

       \[\leadsto \color{blue}{\frac{-\pi}{a \cdot b}} \cdot \frac{0.5}{b - a} \]

   if -2.25e-72 < a < 1.06e-46

   1. Initial program 79.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 79.8%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 79.8%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 79.8%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 87.0%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 87.0%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 87.0%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 87.0%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 87.0%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 87.0%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 87.0%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. inv-pow 87.0%

         \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   5. Applied egg-rr 87.0%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 87.0%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. +-commutative 87.0%

         \[\leadsto \left(\frac{\frac{1}{\frac{\color{blue}{a + b}}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   7. Simplified 87.0%

      \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{a + b}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   8. Taylor expanded in a around 0 82.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 82.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 82.0%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 82.0%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 82.0%

         \[\leadsto \frac{\pi}{a} \cdot \frac{0.5}{\color{blue}{b \cdot b}} \]

   10. Simplified 82.0%

       \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]
   11. Step-by-step derivation

       1. clear-num 81.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{\pi}}} \cdot \frac{0.5}{b \cdot b} \]

       2. associate-/r* 81.9%

          \[\leadsto \frac{1}{\frac{a}{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{b}}{b}} \]

       3. frac-times 94.7%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{0.5}{b}}{\frac{a}{\pi} \cdot b}} \]

       4. *-un-lft-identity 94.7%

          \[\leadsto \frac{\color{blue}{\frac{0.5}{b}}}{\frac{a}{\pi} \cdot b} \]

   12. Applied egg-rr 94.7%

       \[\leadsto \color{blue}{\frac{\frac{0.5}{b}}{\frac{a}{\pi} \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-72} \lor \neg \left(a \leq 1.06 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{-\pi}{b \cdot a} \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{b \cdot \frac{a}{\pi}}\\ \end{array} \]

Alternative 3: 74.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3900000000 \lor \neg \left(b \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -3900000000.0) (not (<= b 9e-58)))
   (* (/ PI a) (/ 0.5 (* b b)))
   (* 0.5 (/ (/ PI b) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -3900000000.0) || !(b <= 9e-58)) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	} else {
		tmp = 0.5 * ((((double) M_PI) / b) / (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -3900000000.0) || !(b <= 9e-58)) {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	} else {
		tmp = 0.5 * ((Math.PI / b) / (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -3900000000.0) or not (b <= 9e-58):
		tmp = (math.pi / a) * (0.5 / (b * b))
	else:
		tmp = 0.5 * ((math.pi / b) / (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -3900000000.0) || !(b <= 9e-58))
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / b) / Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -3900000000.0) || ~((b <= 9e-58)))
		tmp = (pi / a) * (0.5 / (b * b));
	else
		tmp = 0.5 * ((pi / b) / (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -3900000000.0], N[Not[LessEqual[b, 9e-58]], $MachinePrecision]], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9e9 or 9.0000000000000006e-58 < b

   1. Initial program 78.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 78.6%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 78.6%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 90.2%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 90.3%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 90.3%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. inv-pow 90.3%

         \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   5. Applied egg-rr 90.3%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 90.3%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. +-commutative 90.3%

         \[\leadsto \left(\frac{\frac{1}{\frac{\color{blue}{a + b}}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   7. Simplified 90.3%

      \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{a + b}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   8. Taylor expanded in a around 0 86.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 86.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 84.7%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 84.7%

         \[\leadsto \frac{\pi}{a} \cdot \frac{0.5}{\color{blue}{b \cdot b}} \]

   10. Simplified 84.7%

       \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]

   if -3.9e9 < b < 9.0000000000000006e-58

   1. Initial program 81.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 81.7%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 81.7%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 81.7%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 87.3%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 87.6%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. frac-add 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

      2. *-un-lft-identity 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

   5. Applied egg-rr 87.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

      2. neg-mul-1 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

      3. sub-neg 87.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

   7. Simplified 87.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]

   8. Taylor expanded in b around 0 78.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. *-commutative 78.0%

         \[\leadsto 0.5 \cdot \frac{\pi}{\color{blue}{b \cdot {a}^{2}}} \]

      2. associate-/r* 78.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{\pi}{b}}{{a}^{2}}} \]

      3. unpow2 78.0%

         \[\leadsto 0.5 \cdot \frac{\frac{\pi}{b}}{\color{blue}{a \cdot a}} \]

   10. Simplified 78.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3900000000 \lor \neg \left(b \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}\\ \end{array} \]

Alternative 4: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -165000000000 \lor \neg \left(b \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -165000000000.0) (not (<= b 9e-58)))
   (* (/ PI a) (/ 0.5 (* b b)))
   (* 0.5 (/ PI (* a (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -165000000000.0) || !(b <= 9e-58)) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -165000000000.0) || !(b <= 9e-58)) {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -165000000000.0) or not (b <= 9e-58):
		tmp = (math.pi / a) * (0.5 / (b * b))
	else:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -165000000000.0) || !(b <= 9e-58))
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -165000000000.0) || ~((b <= 9e-58)))
		tmp = (pi / a) * (0.5 / (b * b));
	else
		tmp = 0.5 * (pi / (a * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -165000000000.0], N[Not[LessEqual[b, 9e-58]], $MachinePrecision]], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.65e11 or 9.0000000000000006e-58 < b

   1. Initial program 78.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 78.6%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 78.6%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 90.2%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 90.3%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 90.3%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. inv-pow 90.3%

         \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   5. Applied egg-rr 90.3%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 90.3%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. +-commutative 90.3%

         \[\leadsto \left(\frac{\frac{1}{\frac{\color{blue}{a + b}}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   7. Simplified 90.3%

      \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{a + b}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   8. Taylor expanded in a around 0 86.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 86.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 84.7%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 84.7%

         \[\leadsto \frac{\pi}{a} \cdot \frac{0.5}{\color{blue}{b \cdot b}} \]

   10. Simplified 84.7%

       \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]

   if -1.65e11 < b < 9.0000000000000006e-58

   1. Initial program 81.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 81.6%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 81.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 81.7%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 81.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]

   4. Taylor expanded in b around 0 67.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]
   5. Step-by-step derivation

      1. associate-*r/ 67.7%

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]

      2. mul-1-neg 67.7%

         \[\leadsto \frac{\frac{\color{blue}{-\pi}}{b}}{b \cdot b - a \cdot a} \cdot 0.5 \]

   6. Simplified 67.7%

      \[\leadsto \frac{\color{blue}{\frac{-\pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]

   7. Taylor expanded in b around 0 78.0%

      \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. unpow2 78.0%

         \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

      2. associate-*l* 89.9%

         \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   9. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -165000000000 \lor \neg \left(b \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 5: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1400000000 \lor \neg \left(b \leq 8.5 \cdot 10^{-58}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1400000000.0) (not (<= b 8.5e-58)))
   (* 0.5 (/ PI (* a (* b b))))
   (* 0.5 (/ PI (* a (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1400000000.0) || !(b <= 8.5e-58)) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -1400000000.0) || !(b <= 8.5e-58)) {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -1400000000.0) or not (b <= 8.5e-58):
		tmp = 0.5 * (math.pi / (a * (b * b)))
	else:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -1400000000.0) || !(b <= 8.5e-58))
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -1400000000.0) || ~((b <= 8.5e-58)))
		tmp = 0.5 * (pi / (a * (b * b)));
	else
		tmp = 0.5 * (pi / (a * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -1400000000.0], N[Not[LessEqual[b, 8.5e-58]], $MachinePrecision]], N[(0.5 * N[(Pi / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.4e9 or 8.5000000000000004e-58 < b

   1. Initial program 78.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 78.7%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 78.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 78.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 78.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 78.7%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 78.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 78.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]

   4. Taylor expanded in b around inf 86.0%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 86.0%

         \[\leadsto \frac{\pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \cdot 0.5 \]

   6. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(b \cdot b\right)}} \cdot 0.5 \]

   if -1.4e9 < b < 8.5000000000000004e-58

   1. Initial program 81.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 81.6%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 81.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 81.7%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 81.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]

   4. Taylor expanded in b around 0 67.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]
   5. Step-by-step derivation

      1. associate-*r/ 67.7%

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]

      2. mul-1-neg 67.7%

         \[\leadsto \frac{\frac{\color{blue}{-\pi}}{b}}{b \cdot b - a \cdot a} \cdot 0.5 \]

   6. Simplified 67.7%

      \[\leadsto \frac{\color{blue}{\frac{-\pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]

   7. Taylor expanded in b around 0 78.0%

      \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. unpow2 78.0%

         \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

      2. associate-*l* 89.9%

         \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   9. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1400000000 \lor \neg \left(b \leq 8.5 \cdot 10^{-58}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 6: 86.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+14} \lor \neg \left(b \leq 5.4 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{0.5}{b \cdot \left(b \cdot \frac{a}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.02e+14) (not (<= b 5.4e-58)))
   (/ 0.5 (* b (* b (/ a PI))))
   (* 0.5 (/ PI (* a (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1.02e+14) || !(b <= 5.4e-58)) {
		tmp = 0.5 / (b * (b * (a / ((double) M_PI))));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -1.02e+14) || !(b <= 5.4e-58)) {
		tmp = 0.5 / (b * (b * (a / Math.PI)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -1.02e+14) or not (b <= 5.4e-58):
		tmp = 0.5 / (b * (b * (a / math.pi)))
	else:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -1.02e+14) || !(b <= 5.4e-58))
		tmp = Float64(0.5 / Float64(b * Float64(b * Float64(a / pi))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -1.02e+14) || ~((b <= 5.4e-58)))
		tmp = 0.5 / (b * (b * (a / pi)));
	else
		tmp = 0.5 * (pi / (a * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -1.02e+14], N[Not[LessEqual[b, 5.4e-58]], $MachinePrecision]], N[(0.5 / N[(b * N[(b * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.02e14 or 5.3999999999999998e-58 < b

   1. Initial program 78.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 78.6%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 78.6%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 90.2%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 90.3%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 90.3%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 90.3%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. inv-pow 90.3%

         \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   5. Applied egg-rr 90.3%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 90.3%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. +-commutative 90.3%

         \[\leadsto \left(\frac{\frac{1}{\frac{\color{blue}{a + b}}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   7. Simplified 90.3%

      \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{a + b}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   8. Taylor expanded in a around 0 86.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 86.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 86.0%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 84.7%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 84.7%

         \[\leadsto \frac{\pi}{a} \cdot \frac{0.5}{\color{blue}{b \cdot b}} \]

   10. Simplified 84.7%

       \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]
   11. Step-by-step derivation

       1. clear-num 84.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{\pi}}} \cdot \frac{0.5}{b \cdot b} \]

       2. frac-times 86.0%

          \[\leadsto \color{blue}{\frac{1 \cdot 0.5}{\frac{a}{\pi} \cdot \left(b \cdot b\right)}} \]

       3. metadata-eval 86.0%

          \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\pi} \cdot \left(b \cdot b\right)} \]

   12. Applied egg-rr 86.0%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\pi} \cdot \left(b \cdot b\right)}} \]

   13. Taylor expanded in a around 0 85.9%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{a \cdot {b}^{2}}{\pi}}} \]
   14. Step-by-step derivation

       1. *-commutative 85.9%

          \[\leadsto \frac{0.5}{\frac{\color{blue}{{b}^{2} \cdot a}}{\pi}} \]

       2. associate-*r/ 86.0%

          \[\leadsto \frac{0.5}{\color{blue}{{b}^{2} \cdot \frac{a}{\pi}}} \]

       3. unpow2 86.0%

          \[\leadsto \frac{0.5}{\color{blue}{\left(b \cdot b\right)} \cdot \frac{a}{\pi}} \]

       4. associate-*l* 95.2%

          \[\leadsto \frac{0.5}{\color{blue}{b \cdot \left(b \cdot \frac{a}{\pi}\right)}} \]

   15. Simplified 95.2%

       \[\leadsto \frac{0.5}{\color{blue}{b \cdot \left(b \cdot \frac{a}{\pi}\right)}} \]

   if -1.02e14 < b < 5.3999999999999998e-58

   1. Initial program 81.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 81.6%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 81.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 81.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 81.7%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 81.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]

   4. Taylor expanded in b around 0 67.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]
   5. Step-by-step derivation

      1. associate-*r/ 67.7%

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]

      2. mul-1-neg 67.7%

         \[\leadsto \frac{\frac{\color{blue}{-\pi}}{b}}{b \cdot b - a \cdot a} \cdot 0.5 \]

   6. Simplified 67.7%

      \[\leadsto \frac{\color{blue}{\frac{-\pi}{b}}}{b \cdot b - a \cdot a} \cdot 0.5 \]

   7. Taylor expanded in b around 0 78.0%

      \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. unpow2 78.0%

         \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

      2. associate-*l* 89.9%

         \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   9. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+14} \lor \neg \left(b \leq 5.4 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{0.5}{b \cdot \left(b \cdot \frac{a}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 7: 57.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. times-frac 80.1%

      \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. *-commutative 80.1%

      \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. times-frac 80.1%

      \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. difference-of-squares 88.7%

      \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. associate-/r* 89.0%

      \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. metadata-eval 89.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. sub-neg 89.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

   8. distribute-neg-frac 89.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

   9. metadata-eval 89.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Simplified 89.0%

   \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
4. Step-by-step derivation

   1. frac-add 88.9%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

   2. *-un-lft-identity 88.9%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

5. Applied egg-rr 88.9%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
6. Step-by-step derivation

   1. *-commutative 88.9%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

   2. neg-mul-1 88.9%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

   3. sub-neg 88.9%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

7. Simplified 88.9%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]

8. Taylor expanded in b around 0 58.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
9. Step-by-step derivation

   1. *-commutative 58.2%

      \[\leadsto 0.5 \cdot \frac{\pi}{\color{blue}{b \cdot {a}^{2}}} \]

   2. associate-/r* 58.2%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{\pi}{b}}{{a}^{2}}} \]

   3. unpow2 58.2%

      \[\leadsto 0.5 \cdot \frac{\frac{\pi}{b}}{\color{blue}{a \cdot a}} \]

10. Simplified 58.2%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}} \]

11. Final simplification 58.2%

    \[\leadsto 0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a} \]

Reproduce

herbie shell --seed 2023182 
(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))))