NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.7%

Time: 11.0s

Alternatives: 6

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. inv-pow 74.3%

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

   2. difference-of-squares 84.4%

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

   3. unpow-prod-down 85.1%

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

   4. inv-pow 85.1%

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

   5. inv-pow 85.1%

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

3. Applied egg-rr 85.1%

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

   1. associate-*r/ 85.2%

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

   2. *-rgt-identity 85.2%

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

   3. +-commutative 85.2%

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

5. Simplified 85.2%

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

   1. pow1 85.2%

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

   2. frac-times 85.2%

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

   3. div-inv 85.2%

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

   4. inv-pow 85.2%

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

   5. inv-pow 85.2%

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

7. Applied egg-rr 85.2%

   \[\leadsto \color{blue}{{\left(\frac{\frac{\pi}{a + b}}{2 \cdot \left(b - a\right)} \cdot \left({a}^{-1} - {b}^{-1}\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 85.2%

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

   2. associate-*l/ 99.6%

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

   3. times-frac 99.6%

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

   4. unpow-1 99.6%

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

   5. unpow-1 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in a around 0 99.6%

    \[\leadsto \frac{\frac{\pi}{a + b}}{2} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
11. Step-by-step derivation

    1. associate-/r* 99.7%

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

12. Simplified 99.7%

    \[\leadsto \frac{\frac{\pi}{a + b}}{2} \cdot \color{blue}{\frac{\frac{1}{a}}{b}} \]
13. Step-by-step derivation

    1. associate-/r* 99.6%

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

    2. expm1-log1p-u 78.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{a + b}}{2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

    3. expm1-udef 53.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{a + b}}{2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

    4. *-commutative 53.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{a \cdot b} \cdot \frac{\frac{\pi}{a + b}}{2}}\right)} - 1 \]

    5. frac-times 53.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{\pi}{a + b}}{\left(a \cdot b\right) \cdot 2}}\right)} - 1 \]

    6. *-un-lft-identity 53.6%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{\pi}{a + b}}}{\left(a \cdot b\right) \cdot 2}\right)} - 1 \]

14. Applied egg-rr 53.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{a + b}}{\left(a \cdot b\right) \cdot 2}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 78.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{a + b}}{\left(a \cdot b\right) \cdot 2}\right)\right)} \]

    2. expm1-log1p 99.7%

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

    3. associate-/l/ 98.9%

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

    4. associate-*r* 98.9%

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

    5. associate-/r* 99.7%

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

16. Simplified 99.7%

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

17. Final simplification 99.7%

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

Alternative 2: 66.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-119)
   (* (/ PI a) (/ (/ 0.5 a) b))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-119) {
		tmp = (((double) M_PI) / a) * ((0.5 / a) / b);
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-119) {
		tmp = (Math.PI / a) * ((0.5 / a) / b);
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.8e-119:
		tmp = (math.pi / a) * ((0.5 / a) / b)
	else:
		tmp = (math.pi / a) * (0.5 / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-119)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / a) / b));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e-119)
		tmp = (pi / a) * ((0.5 / a) / b);
	else
		tmp = (pi / a) * (0.5 / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e-119], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.80000000000000047e-119

   1. Initial program 72.4%

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

      1. inv-pow 72.4%

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

      2. difference-of-squares 87.8%

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

      3. unpow-prod-down 89.0%

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

      4. inv-pow 89.0%

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

      5. inv-pow 89.0%

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

   3. Applied egg-rr 89.0%

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

      1. associate-*r/ 89.0%

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

      2. *-rgt-identity 89.0%

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

      3. +-commutative 89.0%

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

   5. Simplified 89.0%

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

      1. pow1 89.0%

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

      2. frac-times 89.0%

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

      3. div-inv 89.0%

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

      4. inv-pow 89.0%

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

      5. inv-pow 89.0%

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

   7. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\pi}{a + b}}{2 \cdot \left(b - a\right)} \cdot \left({a}^{-1} - {b}^{-1}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 89.0%

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

      2. associate-*l/ 99.7%

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

      3. times-frac 99.7%

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

      4. unpow-1 99.7%

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

      5. unpow-1 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in a around 0 99.7%

       \[\leadsto \frac{\frac{\pi}{a + b}}{2} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
   11. Step-by-step derivation

       1. associate-/r* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in a around inf 69.4%

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

       1. associate-*r/ 69.4%

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

       2. unpow2 69.4%

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

       3. *-commutative 69.4%

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

       4. associate-*l* 76.2%

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

       5. associate-/r* 76.2%

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

       6. associate-*r/ 76.2%

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

       7. times-frac 76.3%

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

   15. Simplified 76.3%

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

   if -6.80000000000000047e-119 < a

   1. Initial program 75.5%

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

      1. inv-pow 75.5%

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

      2. difference-of-squares 82.4%

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

      3. unpow-prod-down 82.8%

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

      4. inv-pow 82.8%

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

      5. inv-pow 82.8%

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

   3. Applied egg-rr 82.8%

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

      1. associate-*r/ 82.8%

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

      2. *-rgt-identity 82.8%

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

      3. +-commutative 82.8%

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

   5. Simplified 82.8%

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

   6. Taylor expanded in a around 0 58.5%

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

      1. associate-*r/ 58.5%

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

      2. *-commutative 58.5%

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

      3. times-frac 58.5%

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

      4. unpow2 58.5%

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

   8. Simplified 58.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 3: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e-61)
   (* (/ PI a) (/ (/ 0.5 a) b))
   (* (/ PI b) (/ 0.5 (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-61) {
		tmp = (((double) M_PI) / a) * ((0.5 / a) / b);
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-61) {
		tmp = (Math.PI / a) * ((0.5 / a) / b);
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.8e-61:
		tmp = (math.pi / a) * ((0.5 / a) / b)
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.8e-61)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / a) / b));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.8e-61)
		tmp = (pi / a) * ((0.5 / a) / b);
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.8e-61], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.80000000000000007e-61

   1. Initial program 70.9%

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

      1. inv-pow 70.9%

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

      2. difference-of-squares 89.4%

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

      3. unpow-prod-down 89.9%

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

      4. inv-pow 89.9%

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

      5. inv-pow 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. associate-*r/ 90.0%

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

      2. *-rgt-identity 90.0%

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

      3. +-commutative 90.0%

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

   5. Simplified 90.0%

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

      1. pow1 90.0%

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

      2. frac-times 90.0%

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

      3. div-inv 89.9%

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

      4. inv-pow 89.9%

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

      5. inv-pow 89.9%

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

   7. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\pi}{a + b}}{2 \cdot \left(b - a\right)} \cdot \left({a}^{-1} - {b}^{-1}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 89.9%

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

      2. associate-*l/ 99.7%

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

      3. times-frac 99.7%

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

      4. unpow-1 99.7%

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

      5. unpow-1 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in a around 0 99.7%

       \[\leadsto \frac{\frac{\pi}{a + b}}{2} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
   11. Step-by-step derivation

       1. associate-/r* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in a around inf 76.4%

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

       1. associate-*r/ 76.4%

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

       2. unpow2 76.4%

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

       3. *-commutative 76.4%

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

       4. associate-*l* 84.5%

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

       5. associate-/r* 84.5%

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

       6. associate-*r/ 84.5%

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

       7. times-frac 84.6%

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

   15. Simplified 84.6%

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

   if -1.80000000000000007e-61 < a

   1. Initial program 75.9%

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

      1. inv-pow 75.9%

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

      2. difference-of-squares 82.1%

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

      3. unpow-prod-down 82.9%

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

      4. inv-pow 82.9%

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

      5. inv-pow 82.9%

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

   3. Applied egg-rr 82.9%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. +-commutative 82.9%

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

   5. Simplified 82.9%

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

      1. pow1 82.9%

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

      2. frac-times 82.9%

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

      3. div-inv 83.0%

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

      4. inv-pow 83.0%

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

      5. inv-pow 83.0%

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

   7. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\pi}{a + b}}{2 \cdot \left(b - a\right)} \cdot \left({a}^{-1} - {b}^{-1}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 83.0%

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

      2. associate-*l/ 99.6%

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

      3. times-frac 99.6%

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

      4. unpow-1 99.6%

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

      5. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 57.8%

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

       1. associate-*r/ 57.8%

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

       2. unpow2 57.8%

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

       3. associate-*r* 67.8%

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

       4. times-frac 68.7%

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

       5. *-commutative 68.7%

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

   12. Simplified 68.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 4: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e-49)
   (* (/ PI a) (/ (/ 0.5 a) b))
   (/ (/ (* PI 0.5) (* a b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-49) {
		tmp = (((double) M_PI) / a) * ((0.5 / a) / b);
	} else {
		tmp = ((((double) M_PI) * 0.5) / (a * b)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-49) {
		tmp = (Math.PI / a) * ((0.5 / a) / b);
	} else {
		tmp = ((Math.PI * 0.5) / (a * b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.2e-49:
		tmp = (math.pi / a) * ((0.5 / a) / b)
	else:
		tmp = ((math.pi * 0.5) / (a * b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e-49)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / a) / b));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.2e-49)
		tmp = (pi / a) * ((0.5 / a) / b);
	else
		tmp = ((pi * 0.5) / (a * b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e-49], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.19999999999999996e-49

   1. Initial program 69.4%

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

      1. inv-pow 69.4%

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

      2. difference-of-squares 89.6%

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

      3. unpow-prod-down 90.2%

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

      4. inv-pow 90.2%

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

      5. inv-pow 90.2%

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

   3. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.3%

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

      2. *-rgt-identity 90.3%

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

      3. +-commutative 90.3%

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

   5. Simplified 90.3%

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

      1. pow1 90.3%

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

      2. frac-times 90.2%

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

      3. div-inv 90.2%

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

      4. inv-pow 90.2%

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

      5. inv-pow 90.2%

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

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\pi}{a + b}}{2 \cdot \left(b - a\right)} \cdot \left({a}^{-1} - {b}^{-1}\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 90.2%

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

      2. associate-*l/ 99.7%

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

      3. times-frac 99.7%

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

      4. unpow-1 99.7%

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

      5. unpow-1 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in a around 0 99.7%

       \[\leadsto \frac{\frac{\pi}{a + b}}{2} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
   11. Step-by-step derivation

       1. associate-/r* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in a around inf 79.3%

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

       1. associate-*r/ 79.3%

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

       2. unpow2 79.3%

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

       3. *-commutative 79.3%

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

       4. associate-*l* 88.2%

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

       5. associate-/r* 88.2%

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

       6. associate-*r/ 88.2%

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

       7. times-frac 88.3%

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

   15. Simplified 88.3%

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

   if -1.19999999999999996e-49 < a

   1. Initial program 76.3%

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

   2. Taylor expanded in b around inf 57.3%

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

      1. associate-*r/ 57.3%

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

      2. unpow2 57.3%

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

   4. Simplified 57.3%

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

      1. times-frac 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. frac-times 57.3%

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

      2. associate-*l* 67.5%

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

      3. associate-/r* 68.5%

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

      4. *-commutative 68.5%

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

   8. Applied egg-rr 68.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \end{array} \]

Alternative 5: 57.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

2. Taylor expanded in b around inf 54.9%

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

   1. associate-*r/ 54.9%

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

   2. unpow2 54.9%

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

4. Simplified 54.9%

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

   1. times-frac 54.7%

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

6. Applied egg-rr 54.7%

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

7. Final simplification 54.7%

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

Alternative 6: 57.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. inv-pow 74.3%

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

   2. difference-of-squares 84.4%

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

   3. unpow-prod-down 85.1%

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

   4. inv-pow 85.1%

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

   5. inv-pow 85.1%

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

3. Applied egg-rr 85.1%

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

   1. associate-*r/ 85.2%

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

   2. *-rgt-identity 85.2%

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

   3. +-commutative 85.2%

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

5. Simplified 85.2%

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

6. Taylor expanded in a around 0 54.9%

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

   1. associate-*r/ 54.9%

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

   2. *-commutative 54.9%

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

   3. times-frac 54.6%

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

   4. unpow2 54.6%

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

8. Simplified 54.6%

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

9. Final simplification 54.6%

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

Reproduce

herbie shell --seed 2023275 
(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))))