NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 12.0s

Alternatives: 12

Speedup: 1.2×

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

\[\begin{array}{l} \\ \frac{\frac{\pi \cdot 0.5}{b + a} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (* (/ (* PI 0.5) (+ b a)) (+ (/ 1.0 a) (/ -1.0 b))) (- b a)))

C

double code(double a, double b) {
	return (((((double) M_PI) * 0.5) / (b + a)) * ((1.0 / a) + (-1.0 / b))) / (b - a);
}

Java

public static double code(double a, double b) {
	return (((Math.PI * 0.5) / (b + a)) * ((1.0 / a) + (-1.0 / b))) / (b - a);
}

Python

def code(a, b):
	return (((math.pi * 0.5) / (b + a)) * ((1.0 / a) + (-1.0 / b))) / (b - a)

Julia

function code(a, b)
	return Float64(Float64(Float64(Float64(pi * 0.5) / Float64(b + a)) * Float64(Float64(1.0 / a) + Float64(-1.0 / b))) / Float64(b - a))
end

MATLAB

function tmp = code(a, b)
	tmp = (((pi * 0.5) / (b + a)) * ((1.0 / a) + (-1.0 / b))) / (b - a);
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. un-div-inv 78.0%

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

   2. difference-of-squares 89.0%

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

   3. associate-/r* 89.1%

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

   4. div-inv 89.1%

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

   5. metadata-eval 89.1%

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

4. Applied egg-rr 89.1%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{\frac{\pi \cdot 0.5}{b + a} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{b - a} \]
8. Add Preprocessing

Alternative 2: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b} + \frac{-1}{a}}{\left(b + a\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.5e-224)
   (/ (/ (* PI 0.5) (+ b a)) (* b (- a b)))
   (if (<= b 2.2e+68)
     (* (* PI 0.5) (/ (+ (/ 1.0 b) (/ -1.0 a)) (* (+ b a) (- a b))))
     (/ (/ (/ (* PI 0.5) a) b) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-224) {
		tmp = ((((double) M_PI) * 0.5) / (b + a)) / (b * (a - b));
	} else if (b <= 2.2e+68) {
		tmp = (((double) M_PI) * 0.5) * (((1.0 / b) + (-1.0 / a)) / ((b + a) * (a - b)));
	} else {
		tmp = (((((double) M_PI) * 0.5) / a) / b) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-224) {
		tmp = ((Math.PI * 0.5) / (b + a)) / (b * (a - b));
	} else if (b <= 2.2e+68) {
		tmp = (Math.PI * 0.5) * (((1.0 / b) + (-1.0 / a)) / ((b + a) * (a - b)));
	} else {
		tmp = (((Math.PI * 0.5) / a) / b) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.5e-224:
		tmp = ((math.pi * 0.5) / (b + a)) / (b * (a - b))
	elif b <= 2.2e+68:
		tmp = (math.pi * 0.5) * (((1.0 / b) + (-1.0 / a)) / ((b + a) * (a - b)))
	else:
		tmp = (((math.pi * 0.5) / a) / b) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.5e-224)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b + a)) / Float64(b * Float64(a - b)));
	elseif (b <= 2.2e+68)
		tmp = Float64(Float64(pi * 0.5) * Float64(Float64(Float64(1.0 / b) + Float64(-1.0 / a)) / Float64(Float64(b + a) * Float64(a - b))));
	else
		tmp = Float64(Float64(Float64(Float64(pi * 0.5) / a) / b) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.5e-224)
		tmp = ((pi * 0.5) / (b + a)) / (b * (a - b));
	elseif (b <= 2.2e+68)
		tmp = (pi * 0.5) * (((1.0 / b) + (-1.0 / a)) / ((b + a) * (a - b)));
	else
		tmp = (((pi * 0.5) / a) / b) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.5e-224], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e+68], N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(N[(1.0 / b), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.5000000000000003e-224

   1. Initial program 77.7%

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

      1. un-div-inv 77.7%

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

      2. difference-of-squares 88.8%

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

      3. associate-/r* 88.9%

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

      4. div-inv 88.9%

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

      5. metadata-eval 88.9%

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

   4. Applied egg-rr 88.9%

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

   5. Taylor expanded in a around inf 67.8%

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

      1. *-commutative 67.8%

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

      2. frac-2neg 67.8%

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

      3. metadata-eval 67.8%

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

      4. frac-times 76.4%

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

      5. *-un-lft-identity 76.4%

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

   7. Applied egg-rr 76.4%

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

   if 9.5000000000000003e-224 < b < 2.19999999999999987e68

   1. Initial program 95.0%

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

      1. associate-*l* 95.1%

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

      2. *-rgt-identity 95.1%

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

      3. associate-/l* 95.1%

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

      4. metadata-eval 95.1%

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

      5. associate-*l/ 95.1%

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

      6. *-lft-identity 95.1%

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

      7. sub-neg 95.1%

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

      8. distribute-neg-frac 95.1%

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

      9. metadata-eval 95.1%

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

   3. Simplified 95.1%

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

      1. difference-of-squares 95.2%

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

   6. Applied egg-rr 95.2%

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

   if 2.19999999999999987e68 < b

   1. Initial program 51.5%

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

      1. un-div-inv 51.6%

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

      2. difference-of-squares 79.8%

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

      3. associate-/r* 79.8%

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

      4. div-inv 79.8%

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

      5. metadata-eval 79.8%

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

   4. Applied egg-rr 79.8%

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

      1. associate-*l/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around inf 97.6%

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

      1. distribute-lft-out 97.6%

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

      2. sub-neg 97.6%

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

      3. mul-1-neg 97.6%

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

      4. distribute-rgt-out 97.6%

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

      5. metadata-eval 97.6%

         \[\leadsto \frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot \color{blue}{-2}}{b}\right)}{b}}{b - a} \]

   9. Simplified 97.6%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot -2}{b}\right)}{b}}}{b - a} \]

   10. Taylor expanded in a around 0 99.9%

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

       1. associate-*r/ 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b} + \frac{-1}{a}}{\left(b + a\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (* (* PI 0.5) (/ (+ (/ 1.0 a) (/ -1.0 b)) (+ b a))) (- b a)))

C

double code(double a, double b) {
	return ((((double) M_PI) * 0.5) * (((1.0 / a) + (-1.0 / b)) / (b + a))) / (b - a);
}

Java

public static double code(double a, double b) {
	return ((Math.PI * 0.5) * (((1.0 / a) + (-1.0 / b)) / (b + a))) / (b - a);
}

Python

def code(a, b):
	return ((math.pi * 0.5) * (((1.0 / a) + (-1.0 / b)) / (b + a))) / (b - a)

Julia

function code(a, b)
	return Float64(Float64(Float64(pi * 0.5) * Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(b + a))) / Float64(b - a))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi * 0.5) * (((1.0 / a) + (-1.0 / b)) / (b + a))) / (b - a);
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. un-div-inv 78.0%

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

   2. difference-of-squares 89.0%

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

   3. associate-/r* 89.1%

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

   4. div-inv 89.1%

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

   5. metadata-eval 89.1%

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

4. Applied egg-rr 89.1%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-/l* 99.7%

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

   2. *-commutative 99.7%

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

   3. +-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}}{b - a} \]
12. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (/ (* PI 0.5) (+ b a)) (/ (+ (/ 1.0 a) (/ -1.0 b)) (- b a))))

C

double code(double a, double b) {
	return ((((double) M_PI) * 0.5) / (b + a)) * (((1.0 / a) + (-1.0 / b)) / (b - a));
}

Java

public static double code(double a, double b) {
	return ((Math.PI * 0.5) / (b + a)) * (((1.0 / a) + (-1.0 / b)) / (b - a));
}

Python

def code(a, b):
	return ((math.pi * 0.5) / (b + a)) * (((1.0 / a) + (-1.0 / b)) / (b - a))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi * 0.5) / Float64(b + a)) * Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(b - a)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi * 0.5) / (b + a)) * (((1.0 / a) + (-1.0 / b)) / (b - a));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. un-div-inv 78.0%

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

   2. difference-of-squares 89.0%

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

   3. associate-/r* 89.1%

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

   4. div-inv 89.1%

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

   5. metadata-eval 89.1%

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

4. Applied egg-rr 89.1%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \]
10. Add Preprocessing

Alternative 5: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \cdot \left(\pi \cdot \frac{0.5}{b + a}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (/ (+ (/ 1.0 a) (/ -1.0 b)) (- b a)) (* PI (/ 0.5 (+ b a)))))

C

double code(double a, double b) {
	return (((1.0 / a) + (-1.0 / b)) / (b - a)) * (((double) M_PI) * (0.5 / (b + a)));
}

Java

public static double code(double a, double b) {
	return (((1.0 / a) + (-1.0 / b)) / (b - a)) * (Math.PI * (0.5 / (b + a)));
}

Python

def code(a, b):
	return (((1.0 / a) + (-1.0 / b)) / (b - a)) * (math.pi * (0.5 / (b + a)))

Julia

function code(a, b)
	return Float64(Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(b - a)) * Float64(pi * Float64(0.5 / Float64(b + a))))
end

MATLAB

function tmp = code(a, b)
	tmp = (((1.0 / a) + (-1.0 / b)) / (b - a)) * (pi * (0.5 / (b + a)));
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. un-div-inv 78.0%

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

   2. difference-of-squares 89.0%

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

   3. associate-/r* 89.1%

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

   4. div-inv 89.1%

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

   5. metadata-eval 89.1%

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

4. Applied egg-rr 89.1%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. +-commutative 99.6%

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

   4. sub-neg 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \cdot \left(\pi \cdot \frac{0.5}{b + a}\right) \]
10. Add Preprocessing

Alternative 6: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b + a} \cdot \frac{\frac{\pi}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e-175)
   (/ (/ (* PI 0.5) (+ b a)) (* b (- a b)))
   (* (/ 0.5 (+ b a)) (/ (/ PI a) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = ((((double) M_PI) * 0.5) / (b + a)) / (b * (a - b));
	} else {
		tmp = (0.5 / (b + a)) * ((((double) M_PI) / a) / (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = ((Math.PI * 0.5) / (b + a)) / (b * (a - b));
	} else {
		tmp = (0.5 / (b + a)) * ((Math.PI / a) / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.8e-175:
		tmp = ((math.pi * 0.5) / (b + a)) / (b * (a - b))
	else:
		tmp = (0.5 / (b + a)) * ((math.pi / a) / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e-175)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b + a)) / Float64(b * Float64(a - b)));
	else
		tmp = Float64(Float64(0.5 / Float64(b + a)) * Float64(Float64(pi / a) / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.8e-175)
		tmp = ((pi * 0.5) / (b + a)) / (b * (a - b));
	else
		tmp = (0.5 / (b + a)) * ((pi / a) / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e-175], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.79999999999999998e-175

   1. Initial program 78.0%

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

      1. un-div-inv 78.0%

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

      2. difference-of-squares 88.3%

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

      3. associate-/r* 88.5%

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

      4. div-inv 88.5%

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

      5. metadata-eval 88.5%

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

   4. Applied egg-rr 88.5%

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

   5. Taylor expanded in a around inf 68.2%

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

      1. *-commutative 68.2%

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

      2. frac-2neg 68.2%

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

      3. metadata-eval 68.2%

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

      4. frac-times 77.4%

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

      5. *-un-lft-identity 77.4%

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

   7. Applied egg-rr 77.4%

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

   if 5.79999999999999998e-175 < b

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.1%

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

      3. associate-*r/ 78.2%

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

      4. associate-*r* 78.2%

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

      5. *-rgt-identity 78.2%

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

      6. sub-neg 78.2%

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

      7. distribute-neg-frac 78.2%

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

      8. metadata-eval 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in a around 0 62.9%

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

      1. difference-of-squares 90.1%

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

   7. Applied egg-rr 74.9%

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

      1. times-frac 83.4%

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b + a} \cdot \frac{\frac{\pi}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{-0.5}{b + a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b + a} \cdot \frac{\frac{\pi}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e-175)
   (* (/ -0.5 (+ b a)) (/ (/ PI b) (- b a)))
   (* (/ 0.5 (+ b a)) (/ (/ PI a) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = (-0.5 / (b + a)) * ((((double) M_PI) / b) / (b - a));
	} else {
		tmp = (0.5 / (b + a)) * ((((double) M_PI) / a) / (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = (-0.5 / (b + a)) * ((Math.PI / b) / (b - a));
	} else {
		tmp = (0.5 / (b + a)) * ((Math.PI / a) / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.8e-175:
		tmp = (-0.5 / (b + a)) * ((math.pi / b) / (b - a))
	else:
		tmp = (0.5 / (b + a)) * ((math.pi / a) / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e-175)
		tmp = Float64(Float64(-0.5 / Float64(b + a)) * Float64(Float64(pi / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 / Float64(b + a)) * Float64(Float64(pi / a) / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.8e-175)
		tmp = (-0.5 / (b + a)) * ((pi / b) / (b - a));
	else
		tmp = (0.5 / (b + a)) * ((pi / a) / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e-175], N[(N[(-0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.79999999999999998e-175

   1. Initial program 78.0%

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

      1. un-div-inv 78.0%

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

      2. difference-of-squares 88.3%

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

      3. associate-/r* 88.5%

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

      4. div-inv 88.5%

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

      5. metadata-eval 88.5%

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

   4. Applied egg-rr 88.5%

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

   5. Taylor expanded in a around inf 68.2%

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

      1. *-commutative 68.2%

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

      2. frac-2neg 68.2%

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

      3. metadata-eval 68.2%

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

      4. associate-/l/ 68.2%

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

      5. *-commutative 68.2%

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

      6. frac-times 68.2%

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

      7. *-un-lft-identity 68.2%

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

   7. Applied egg-rr 68.2%

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

      1. associate-/r* 68.3%

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

      2. distribute-neg-frac2 68.3%

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

      3. distribute-neg-frac 68.3%

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

      4. distribute-rgt-neg-in 68.3%

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

      5. metadata-eval 68.3%

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

      6. *-commutative 68.3%

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

      7. associate-*r/ 68.3%

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

      8. +-commutative 68.3%

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

   9. Simplified 68.3%

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

       1. times-frac 77.3%

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

   11. Applied egg-rr 77.3%

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

   if 5.79999999999999998e-175 < b

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.1%

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

      3. associate-*r/ 78.2%

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

      4. associate-*r* 78.2%

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

      5. *-rgt-identity 78.2%

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

      6. sub-neg 78.2%

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

      7. distribute-neg-frac 78.2%

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

      8. metadata-eval 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in a around 0 62.9%

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

      1. difference-of-squares 90.1%

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

   7. Applied egg-rr 74.9%

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

      1. times-frac 83.4%

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{-0.5}{b + a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b + a} \cdot \frac{\frac{\pi}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5}{b + a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.6e-105)
   (* (/ -0.5 (+ b a)) (/ (/ PI b) (- b a)))
   (/ (/ (/ (* PI 0.5) a) b) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.6e-105) {
		tmp = (-0.5 / (b + a)) * ((((double) M_PI) / b) / (b - a));
	} else {
		tmp = (((((double) M_PI) * 0.5) / a) / b) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.6e-105) {
		tmp = (-0.5 / (b + a)) * ((Math.PI / b) / (b - a));
	} else {
		tmp = (((Math.PI * 0.5) / a) / b) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.6e-105:
		tmp = (-0.5 / (b + a)) * ((math.pi / b) / (b - a))
	else:
		tmp = (((math.pi * 0.5) / a) / b) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.6e-105)
		tmp = Float64(Float64(-0.5 / Float64(b + a)) * Float64(Float64(pi / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(Float64(pi * 0.5) / a) / b) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.6e-105)
		tmp = (-0.5 / (b + a)) * ((pi / b) / (b - a));
	else
		tmp = (((pi * 0.5) / a) / b) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.6e-105], N[(N[(-0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.59999999999999964e-105

   1. Initial program 79.0%

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

      1. un-div-inv 79.0%

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

      2. difference-of-squares 88.9%

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

      3. associate-/r* 89.0%

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

      4. div-inv 89.0%

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

      5. metadata-eval 89.0%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in a around inf 69.7%

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

      1. *-commutative 69.7%

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

      2. frac-2neg 69.7%

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

      3. metadata-eval 69.7%

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

      4. associate-/l/ 69.7%

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

      5. *-commutative 69.7%

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

      6. frac-times 69.7%

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

      7. *-un-lft-identity 69.7%

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

   7. Applied egg-rr 69.7%

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

      1. associate-/r* 69.7%

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

      2. distribute-neg-frac2 69.7%

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

      3. distribute-neg-frac 69.7%

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

      4. distribute-rgt-neg-in 69.7%

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

      5. metadata-eval 69.7%

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

      6. *-commutative 69.7%

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

      7. associate-*r/ 69.7%

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

      8. +-commutative 69.7%

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

   9. Simplified 69.7%

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

       1. times-frac 78.4%

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

   11. Applied egg-rr 78.4%

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

   if 3.59999999999999964e-105 < b

   1. Initial program 76.0%

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

      1. un-div-inv 76.1%

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

      2. difference-of-squares 89.2%

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

      3. associate-/r* 89.3%

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

      4. div-inv 89.3%

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

      5. metadata-eval 89.3%

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

   4. Applied egg-rr 89.3%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf 75.1%

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

      1. distribute-lft-out 75.1%

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

      2. sub-neg 75.1%

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

      3. mul-1-neg 75.1%

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

      4. distribute-rgt-out 75.1%

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

      5. metadata-eval 75.1%

         \[\leadsto \frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot \color{blue}{-2}}{b}\right)}{b}}{b - a} \]

   9. Simplified 75.1%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot -2}{b}\right)}{b}}}{b - a} \]

   10. Taylor expanded in a around 0 87.3%

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

       1. associate-*r/ 87.3%

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

   12. Simplified 87.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5}{b + a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e-175)
   (* -0.5 (/ (/ (/ PI a) b) (- b a)))
   (/ (/ (/ (* PI 0.5) a) b) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = -0.5 * (((((double) M_PI) / a) / b) / (b - a));
	} else {
		tmp = (((((double) M_PI) * 0.5) / a) / b) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = -0.5 * (((Math.PI / a) / b) / (b - a));
	} else {
		tmp = (((Math.PI * 0.5) / a) / b) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.8e-175:
		tmp = -0.5 * (((math.pi / a) / b) / (b - a))
	else:
		tmp = (((math.pi * 0.5) / a) / b) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e-175)
		tmp = Float64(-0.5 * Float64(Float64(Float64(pi / a) / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(Float64(pi * 0.5) / a) / b) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.8e-175)
		tmp = -0.5 * (((pi / a) / b) / (b - a));
	else
		tmp = (((pi * 0.5) / a) / b) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e-175], N[(-0.5 * N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.79999999999999998e-175

   1. Initial program 78.0%

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

      1. un-div-inv 78.0%

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

      2. difference-of-squares 88.3%

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

      3. associate-/r* 88.5%

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

      4. div-inv 88.5%

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

      5. metadata-eval 88.5%

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

   4. Applied egg-rr 88.5%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 72.5%

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

      1. associate-/l* 72.5%

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

      2. associate-/r* 72.6%

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

   9. Applied egg-rr 72.6%

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

   if 5.79999999999999998e-175 < b

   1. Initial program 78.1%

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

      1. un-div-inv 78.1%

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

      2. difference-of-squares 90.1%

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

      3. associate-/r* 90.2%

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

      4. div-inv 90.2%

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

      5. metadata-eval 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf 72.0%

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

      1. distribute-lft-out 72.0%

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

      2. sub-neg 72.0%

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

      3. mul-1-neg 72.0%

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

      4. distribute-rgt-out 72.0%

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

      5. metadata-eval 72.0%

         \[\leadsto \frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot \color{blue}{-2}}{b}\right)}{b}}{b - a} \]

   9. Simplified 72.0%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot -2}{b}\right)}{b}}}{b - a} \]

   10. Taylor expanded in a around 0 81.9%

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

       1. associate-*r/ 81.9%

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

   12. Simplified 81.9%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e-175)
   (* -0.5 (/ (/ (/ PI a) b) (- b a)))
   (/ (* (/ PI b) (/ 0.5 a)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = -0.5 * (((((double) M_PI) / a) / b) / (b - a));
	} else {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.8e-175) {
		tmp = -0.5 * (((Math.PI / a) / b) / (b - a));
	} else {
		tmp = ((Math.PI / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.8e-175:
		tmp = -0.5 * (((math.pi / a) / b) / (b - a))
	else:
		tmp = ((math.pi / b) * (0.5 / a)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e-175)
		tmp = Float64(-0.5 * Float64(Float64(Float64(pi / a) / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.8e-175)
		tmp = -0.5 * (((pi / a) / b) / (b - a));
	else
		tmp = ((pi / b) * (0.5 / a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e-175], N[(-0.5 * N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.79999999999999998e-175

   1. Initial program 78.0%

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

      1. un-div-inv 78.0%

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

      2. difference-of-squares 88.3%

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

      3. associate-/r* 88.5%

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

      4. div-inv 88.5%

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

      5. metadata-eval 88.5%

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

   4. Applied egg-rr 88.5%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 72.5%

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

      1. associate-/l* 72.5%

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

      2. associate-/r* 72.6%

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

   9. Applied egg-rr 72.6%

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

   if 5.79999999999999998e-175 < b

   1. Initial program 78.1%

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

      1. un-div-inv 78.1%

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

      2. difference-of-squares 90.1%

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

      3. associate-/r* 90.2%

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

      4. div-inv 90.2%

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

      5. metadata-eval 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-/l* 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in a around 0 81.9%

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

       1. associate-*r/ 81.9%

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

       2. *-commutative 81.9%

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

       3. *-commutative 81.9%

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

       4. times-frac 81.9%

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

   13. Simplified 81.9%

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

Alternative 11: 51.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-195}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{b}}{a - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.5e-195)
   (* -0.5 (/ (/ (/ PI a) b) (- b a)))
   (/ (/ (/ PI b) b) (- a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-195) {
		tmp = -0.5 * (((((double) M_PI) / a) / b) / (b - a));
	} else {
		tmp = ((((double) M_PI) / b) / b) / (a - b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-195) {
		tmp = -0.5 * (((Math.PI / a) / b) / (b - a));
	} else {
		tmp = ((Math.PI / b) / b) / (a - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.5e-195:
		tmp = -0.5 * (((math.pi / a) / b) / (b - a))
	else:
		tmp = ((math.pi / b) / b) / (a - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.5e-195)
		tmp = Float64(-0.5 * Float64(Float64(Float64(pi / a) / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi / b) / b) / Float64(a - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.5e-195)
		tmp = -0.5 * (((pi / a) / b) / (b - a));
	else
		tmp = ((pi / b) / b) / (a - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.5e-195], N[(-0.5 * N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / b), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.5000000000000003e-195

   1. Initial program 74.9%

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

      1. un-div-inv 74.9%

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

      2. difference-of-squares 84.7%

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

      3. associate-/r* 84.9%

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

      4. div-inv 84.9%

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

      5. metadata-eval 84.9%

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

   4. Applied egg-rr 84.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 77.5%

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

      1. associate-/l* 77.5%

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

      2. associate-/r* 77.6%

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

   9. Applied egg-rr 77.6%

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

   if -5.5000000000000003e-195 < a

   1. Initial program 80.5%

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

      1. un-div-inv 80.5%

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

      2. difference-of-squares 92.3%

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

      3. associate-/r* 92.4%

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

      4. div-inv 92.4%

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

      5. metadata-eval 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf 66.3%

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

      1. distribute-lft-out 66.3%

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

      2. sub-neg 66.3%

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

      3. mul-1-neg 66.3%

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

      4. distribute-rgt-out 66.3%

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

      5. metadata-eval 66.3%

         \[\leadsto \frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot \color{blue}{-2}}{b}\right)}{b}}{b - a} \]

   9. Simplified 66.3%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot -2}{b}\right)}{b}}}{b - a} \]

   10. Taylor expanded in a around inf 31.8%

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

       1. associate-*r/ 31.8%

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

       2. mul-1-neg 31.8%

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

   12. Simplified 31.8%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-195}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{b}}{a - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. un-div-inv 78.0%

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

   2. difference-of-squares 89.0%

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

   3. associate-/r* 89.1%

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

   4. div-inv 89.1%

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

   5. metadata-eval 89.1%

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

4. Applied egg-rr 89.1%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in b around inf 60.6%

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

   1. distribute-lft-out 60.6%

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

   2. sub-neg 60.6%

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

   3. mul-1-neg 60.6%

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

   4. distribute-rgt-out 60.6%

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

   5. metadata-eval 60.6%

      \[\leadsto \frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot \color{blue}{-2}}{b}\right)}{b}}{b - a} \]

9. Simplified 60.6%

   \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot -2}{b}\right)}{b}}}{b - a} \]

10. Taylor expanded in a around inf 31.9%

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

    1. associate-*r/ 31.9%

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

    2. mul-1-neg 31.9%

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

12. Simplified 31.9%

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

13. Final simplification 31.9%

    \[\leadsto \frac{\frac{\frac{\pi}{b}}{b}}{a - b} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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))))