NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 10.0s

Alternatives: 9

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-*l/ 78.4%

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

   3. associate-*r/ 78.4%

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

   4. associate-/l* 78.4%

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

   5. sub-neg 78.4%

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

   6. distribute-neg-frac 78.4%

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

   7. metadata-eval 78.4%

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

   8. associate-*r/ 78.3%

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

   9. *-rgt-identity 78.3%

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

   10. difference-of-squares 87.3%

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

   11. associate-/r* 87.3%

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

3. Simplified 87.3%

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

   1. clear-num 87.2%

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

   2. inv-pow 87.2%

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

   3. associate-/r/ 87.2%

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

5. Applied egg-rr 87.2%

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

   1. unpow-1 87.2%

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

   2. associate-*l/ 87.3%

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

   3. +-commutative 87.3%

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

   4. metadata-eval 87.3%

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

   5. distribute-neg-frac 87.3%

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

   6. sub-neg 87.3%

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

7. Simplified 87.3%

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

   1. *-un-lft-identity 87.3%

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

   2. associate-/l/ 99.2%

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

   3. inv-pow 99.2%

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

   4. inv-pow 99.2%

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

9. Applied egg-rr 99.2%

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

    1. expm1-log1p-u 79.0%

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

    2. expm1-udef 52.4%

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

    3. *-un-lft-identity 52.4%

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

    4. inv-pow 52.4%

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

    5. unpow-1 52.4%

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

    6. +-commutative 52.4%

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

11. Applied egg-rr 52.4%

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

    1. expm1-def 79.0%

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

    2. expm1-log1p 99.2%

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

    3. associate-/r/ 99.6%

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

    4. +-commutative 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-167}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+140}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b - a\right)} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.2e-167)
   (* 0.5 (/ (/ PI a) (* b a)))
   (if (<= b 4.8e+140)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (/ (/ PI 2.0) (- (* b b) (* a a))))
     (* (/ 1.0 (* 2.0 (- b a))) (/ PI (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.2e-167) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * a));
	} else if (b <= 4.8e+140) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((((double) M_PI) / 2.0) / ((b * b) - (a * a)));
	} else {
		tmp = (1.0 / (2.0 * (b - a))) * (((double) M_PI) / (b * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8.2e-167) {
		tmp = 0.5 * ((Math.PI / a) / (b * a));
	} else if (b <= 4.8e+140) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((Math.PI / 2.0) / ((b * b) - (a * a)));
	} else {
		tmp = (1.0 / (2.0 * (b - a))) * (Math.PI / (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8.2e-167:
		tmp = 0.5 * ((math.pi / a) / (b * a))
	elif b <= 4.8e+140:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((math.pi / 2.0) / ((b * b) - (a * a)))
	else:
		tmp = (1.0 / (2.0 * (b - a))) * (math.pi / (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.2e-167)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * a)));
	elseif (b <= 4.8e+140)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(pi / 2.0) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(1.0 / Float64(2.0 * Float64(b - a))) * Float64(pi / Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.2e-167)
		tmp = 0.5 * ((pi / a) / (b * a));
	elseif (b <= 4.8e+140)
		tmp = ((1.0 / a) + (-1.0 / b)) * ((pi / 2.0) / ((b * b) - (a * a)));
	else
		tmp = (1.0 / (2.0 * (b - a))) * (pi / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.2e-167], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e+140], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / 2.0), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8.20000000000000036e-167

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*l/ 77.5%

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

      3. associate-*r/ 77.5%

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

      4. associate-/l* 77.4%

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

      5. sub-neg 77.4%

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

      6. distribute-neg-frac 77.4%

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

      7. metadata-eval 77.4%

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

      8. associate-*r/ 77.4%

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

      9. *-rgt-identity 77.4%

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

      10. difference-of-squares 85.6%

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

      11. associate-/r* 85.6%

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

   3. Simplified 85.6%

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

      1. clear-num 85.6%

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

      2. inv-pow 85.6%

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

      3. associate-/r/ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. unpow-1 85.6%

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

      2. associate-*l/ 85.6%

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

      3. +-commutative 85.6%

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

      4. metadata-eval 85.6%

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

      5. distribute-neg-frac 85.6%

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

      6. sub-neg 85.6%

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

   7. Simplified 85.6%

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

      1. *-un-lft-identity 85.6%

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

      2. associate-/l/ 99.3%

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

      3. inv-pow 99.3%

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

      4. inv-pow 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in b around 0 56.3%

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

       1. *-commutative 56.3%

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

       2. unpow2 56.3%

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

       3. associate-*r* 67.3%

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

       4. *-commutative 67.3%

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

       5. associate-/r* 67.1%

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

       6. *-commutative 67.1%

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

   12. Simplified 67.1%

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

   if 8.20000000000000036e-167 < b < 4.7999999999999999e140

   1. Initial program 99.4%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-neg-frac 99.6%

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

      5. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   if 4.7999999999999999e140 < b

   1. Initial program 49.3%

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

      1. *-commutative 49.3%

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

      2. associate-*l/ 49.3%

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

      3. associate-*r/ 49.3%

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

      4. associate-/l* 49.3%

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

      5. sub-neg 49.3%

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

      6. distribute-neg-frac 49.3%

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

      7. metadata-eval 49.3%

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

      8. associate-*r/ 49.3%

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

      9. *-rgt-identity 49.3%

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

      10. difference-of-squares 76.6%

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

      11. associate-/r* 76.6%

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

   3. Simplified 76.6%

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

      1. clear-num 76.5%

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

      2. inv-pow 76.5%

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

      3. associate-/r/ 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. unpow-1 76.6%

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

      2. associate-*l/ 76.6%

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

      3. +-commutative 76.6%

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

      4. metadata-eval 76.6%

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

      5. distribute-neg-frac 76.6%

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

      6. sub-neg 76.6%

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

   7. Simplified 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. associate-/l/ 98.7%

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

      3. inv-pow 98.7%

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

      4. inv-pow 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. expm1-log1p-u 98.7%

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

       2. expm1-udef 74.0%

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

       3. *-un-lft-identity 74.0%

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

       4. inv-pow 74.0%

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

       5. unpow-1 74.0%

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

       6. +-commutative 74.0%

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

   11. Applied egg-rr 74.0%

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

       1. expm1-def 98.7%

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

       2. expm1-log1p 98.7%

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

       3. associate-/r/ 99.9%

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

       4. +-commutative 99.9%

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

   13. Simplified 99.9%

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

   14. Taylor expanded in a around 0 99.9%

       \[\leadsto \frac{1}{2 \cdot \left(b - a\right)} \cdot \color{blue}{\frac{\pi}{a \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-167}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+140}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b - a\right)} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]

Alternative 3: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{b + a}\\ t_1 := 2 \cdot \left(b - a\right)\\ \mathbf{if}\;a \leq -4 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-190}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{t_0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1} \cdot \left(\frac{1}{a} \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (+ b a))) (t_1 (* 2.0 (- b a))))
   (if (<= a -4e+107)
     (* 0.5 (/ (/ PI a) (* b a)))
     (if (<= a -5.2e-190)
       (* (+ (/ 1.0 a) (/ -1.0 b)) (/ t_0 t_1))
       (* (/ 1.0 t_1) (* (/ 1.0 a) t_0))))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) / (b + a);
	double t_1 = 2.0 * (b - a);
	double tmp;
	if (a <= -4e+107) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * a));
	} else if (a <= -5.2e-190) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / t_1);
	} else {
		tmp = (1.0 / t_1) * ((1.0 / a) * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI / (b + a);
	double t_1 = 2.0 * (b - a);
	double tmp;
	if (a <= -4e+107) {
		tmp = 0.5 * ((Math.PI / a) / (b * a));
	} else if (a <= -5.2e-190) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / t_1);
	} else {
		tmp = (1.0 / t_1) * ((1.0 / a) * t_0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi / (b + a)
	t_1 = 2.0 * (b - a)
	tmp = 0
	if a <= -4e+107:
		tmp = 0.5 * ((math.pi / a) / (b * a))
	elif a <= -5.2e-190:
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / t_1)
	else:
		tmp = (1.0 / t_1) * ((1.0 / a) * t_0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi / Float64(b + a))
	t_1 = Float64(2.0 * Float64(b - a))
	tmp = 0.0
	if (a <= -4e+107)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * a)));
	elseif (a <= -5.2e-190)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(t_0 / t_1));
	else
		tmp = Float64(Float64(1.0 / t_1) * Float64(Float64(1.0 / a) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi / (b + a);
	t_1 = 2.0 * (b - a);
	tmp = 0.0;
	if (a <= -4e+107)
		tmp = 0.5 * ((pi / a) / (b * a));
	elseif (a <= -5.2e-190)
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / t_1);
	else
		tmp = (1.0 / t_1) * ((1.0 / a) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4e+107], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -5.2e-190], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.9999999999999999e107

   1. Initial program 56.4%

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

      1. *-commutative 56.4%

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

      2. associate-*l/ 56.4%

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

      3. associate-*r/ 56.4%

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

      4. associate-/l* 56.4%

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

      5. sub-neg 56.4%

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

      6. distribute-neg-frac 56.4%

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

      7. metadata-eval 56.4%

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

      8. associate-*r/ 56.4%

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

      9. *-rgt-identity 56.4%

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

      10. difference-of-squares 68.3%

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

      11. associate-/r* 68.3%

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

   3. Simplified 68.3%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. associate-/r/ 67.9%

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

   5. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-*l/ 67.9%

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

      3. +-commutative 67.9%

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

      4. metadata-eval 67.9%

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

      5. distribute-neg-frac 67.9%

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

      6. sub-neg 67.9%

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

   7. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-/l/ 99.2%

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

      3. inv-pow 99.2%

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

      4. inv-pow 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in b around 0 68.1%

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

       1. *-commutative 68.1%

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

       2. unpow2 68.1%

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

       3. associate-*r* 99.4%

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

       4. *-commutative 99.4%

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

       5. associate-/r* 99.6%

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

       6. *-commutative 99.6%

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

   12. Simplified 99.6%

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

   if -3.9999999999999999e107 < a < -5.1999999999999996e-190

   1. Initial program 91.9%

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

      1. inv-pow 91.9%

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

      2. difference-of-squares 94.9%

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

      3. unpow-prod-down 94.7%

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

      4. inv-pow 94.7%

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

      5. inv-pow 94.7%

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

   3. Applied egg-rr 94.7%

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

      1. associate-*r/ 94.9%

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

      2. *-rgt-identity 94.9%

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

      3. +-commutative 94.9%

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

   5. Simplified 94.9%

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

      1. frac-times 94.9%

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

      2. div-inv 94.9%

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

   7. Applied egg-rr 94.9%

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

   if -5.1999999999999996e-190 < a

   1. Initial program 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*l/ 78.6%

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

      3. associate-*r/ 78.6%

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

      4. associate-/l* 78.6%

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

      5. sub-neg 78.6%

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

      6. distribute-neg-frac 78.6%

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

      7. metadata-eval 78.6%

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

      8. associate-*r/ 78.6%

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

      9. *-rgt-identity 78.6%

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

      10. difference-of-squares 89.4%

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

      11. associate-/r* 89.3%

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

   3. Simplified 89.3%

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

      1. clear-num 89.4%

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

      2. inv-pow 89.4%

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

      3. associate-/r/ 89.4%

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

   5. Applied egg-rr 89.4%

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

      1. unpow-1 89.4%

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

      2. associate-*l/ 89.4%

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

      3. +-commutative 89.4%

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

      4. metadata-eval 89.4%

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

      5. distribute-neg-frac 89.4%

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

      6. sub-neg 89.4%

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

   7. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. associate-/l/ 99.4%

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

      3. inv-pow 99.4%

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

      4. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. expm1-log1p-u 79.5%

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

       2. expm1-udef 56.9%

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

       3. *-un-lft-identity 56.9%

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

       4. inv-pow 56.9%

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

       5. unpow-1 56.9%

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

       6. +-commutative 56.9%

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

   11. Applied egg-rr 56.9%

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

       1. expm1-def 79.5%

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

       2. expm1-log1p 99.4%

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

       3. associate-/r/ 99.6%

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

       4. +-commutative 99.6%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in a around 0 76.9%

       \[\leadsto \frac{1}{2 \cdot \left(b - a\right)} \cdot \left(\color{blue}{\frac{1}{a}} \cdot \frac{\pi}{a + b}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-190}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{b + a}}{2 \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} \cdot \frac{\pi}{b + a}\right)\\ \end{array} \]

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} \cdot \frac{\pi}{b + a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25e-72)
   (* 0.5 (/ (/ PI a) (* b a)))
   (* (/ 1.0 (* 2.0 (- b a))) (* (/ 1.0 a) (/ PI (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-72) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * a));
	} else {
		tmp = (1.0 / (2.0 * (b - a))) * ((1.0 / a) * (((double) M_PI) / (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-72) {
		tmp = 0.5 * ((Math.PI / a) / (b * a));
	} else {
		tmp = (1.0 / (2.0 * (b - a))) * ((1.0 / a) * (Math.PI / (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.25e-72:
		tmp = 0.5 * ((math.pi / a) / (b * a))
	else:
		tmp = (1.0 / (2.0 * (b - a))) * ((1.0 / a) * (math.pi / (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25e-72)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * a)));
	else
		tmp = Float64(Float64(1.0 / Float64(2.0 * Float64(b - a))) * Float64(Float64(1.0 / a) * Float64(pi / Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.25e-72)
		tmp = 0.5 * ((pi / a) / (b * a));
	else
		tmp = (1.0 / (2.0 * (b - a))) * ((1.0 / a) * (pi / (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25e-72], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] * N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2499999999999999e-72

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l/ 79.6%

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

      3. associate-*r/ 79.6%

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

      4. associate-/l* 79.5%

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

      5. sub-neg 79.5%

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

      6. distribute-neg-frac 79.5%

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

      7. metadata-eval 79.5%

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

      8. associate-*r/ 79.5%

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

      9. *-rgt-identity 79.5%

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

      10. difference-of-squares 86.9%

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

      11. associate-/r* 86.9%

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

   3. Simplified 86.9%

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

      1. clear-num 86.9%

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

      2. inv-pow 86.9%

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

      3. associate-/r/ 86.9%

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

   5. Applied egg-rr 86.9%

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

      1. unpow-1 86.9%

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

      2. associate-*l/ 87.0%

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

      3. +-commutative 87.0%

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

      4. metadata-eval 87.0%

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

      5. distribute-neg-frac 87.0%

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

      6. sub-neg 87.0%

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

   7. Simplified 87.0%

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

      1. *-un-lft-identity 87.0%

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

      2. associate-/l/ 99.4%

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

      3. inv-pow 99.4%

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

      4. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in b around 0 59.4%

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

       1. *-commutative 59.4%

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

       2. unpow2 59.4%

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

       3. associate-*r* 69.4%

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

       4. *-commutative 69.4%

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

       5. associate-/r* 69.2%

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

       6. *-commutative 69.2%

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

   12. Simplified 69.2%

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

   if 1.2499999999999999e-72 < b

   1. Initial program 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-*l/ 75.1%

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

      3. associate-*r/ 75.1%

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

      4. associate-/l* 75.1%

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

      5. sub-neg 75.1%

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

      6. distribute-neg-frac 75.1%

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

      7. metadata-eval 75.1%

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

      8. associate-*r/ 75.1%

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

      9. *-rgt-identity 75.1%

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

      10. difference-of-squares 88.4%

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

      11. associate-/r* 88.3%

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

   3. Simplified 88.3%

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

      1. clear-num 88.1%

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

      2. inv-pow 88.1%

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

      3. associate-/r/ 88.1%

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

   5. Applied egg-rr 88.1%

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

      1. unpow-1 88.1%

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

      2. associate-*l/ 88.2%

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

      3. +-commutative 88.2%

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

      4. metadata-eval 88.2%

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

      5. distribute-neg-frac 88.2%

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

      6. sub-neg 88.2%

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

   7. Simplified 88.2%

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

      1. *-un-lft-identity 88.2%

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

      2. associate-/l/ 98.8%

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

      3. inv-pow 98.8%

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

      4. inv-pow 98.8%

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

   9. Applied egg-rr 98.8%

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

       1. expm1-log1p-u 85.0%

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

       2. expm1-udef 58.6%

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

       3. *-un-lft-identity 58.6%

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

       4. inv-pow 58.6%

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

       5. unpow-1 58.6%

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

       6. +-commutative 58.6%

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

   11. Applied egg-rr 58.6%

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

       1. expm1-def 85.0%

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

       2. expm1-log1p 98.8%

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

       3. associate-/r/ 99.6%

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

       4. +-commutative 99.6%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in a around 0 94.1%

       \[\leadsto \frac{1}{2 \cdot \left(b - a\right)} \cdot \left(\color{blue}{\frac{1}{a}} \cdot \frac{\pi}{a + b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} \cdot \frac{\pi}{b + a}\right)\\ \end{array} \]

Alternative 5: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b - a\right)} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.4e-24)
   (* 0.5 (/ (/ PI a) (* b a)))
   (* (/ 1.0 (* 2.0 (- b a))) (/ PI (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.4e-24) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * a));
	} else {
		tmp = (1.0 / (2.0 * (b - a))) * (((double) M_PI) / (b * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.4e-24) {
		tmp = 0.5 * ((Math.PI / a) / (b * a));
	} else {
		tmp = (1.0 / (2.0 * (b - a))) * (Math.PI / (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.4e-24:
		tmp = 0.5 * ((math.pi / a) / (b * a))
	else:
		tmp = (1.0 / (2.0 * (b - a))) * (math.pi / (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.4e-24)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * a)));
	else
		tmp = Float64(Float64(1.0 / Float64(2.0 * Float64(b - a))) * Float64(pi / Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.4e-24)
		tmp = 0.5 * ((pi / a) / (b * a));
	else
		tmp = (1.0 / (2.0 * (b - a))) * (pi / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.4e-24], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.39999999999999992e-24

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l/ 79.9%

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

      3. associate-*r/ 79.9%

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

      4. associate-/l* 79.9%

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

      5. sub-neg 79.9%

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

      6. distribute-neg-frac 79.9%

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

      7. metadata-eval 79.9%

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

      8. associate-*r/ 79.8%

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

      9. *-rgt-identity 79.8%

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

      10. difference-of-squares 87.2%

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

      11. associate-/r* 87.1%

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

   3. Simplified 87.1%

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

      1. clear-num 87.1%

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

      2. inv-pow 87.1%

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

      3. associate-/r/ 87.1%

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

   5. Applied egg-rr 87.1%

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

      1. unpow-1 87.1%

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

      2. associate-*l/ 87.2%

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

      3. +-commutative 87.2%

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

      4. metadata-eval 87.2%

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

      5. distribute-neg-frac 87.2%

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

      6. sub-neg 87.2%

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

   7. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. associate-/l/ 99.4%

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

      3. inv-pow 99.4%

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

      4. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in b around 0 59.5%

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

       1. *-commutative 59.5%

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

       2. unpow2 59.5%

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

       3. associate-*r* 69.3%

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

       4. *-commutative 69.3%

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

       5. associate-/r* 69.1%

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

       6. *-commutative 69.1%

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

   12. Simplified 69.1%

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

   if 3.39999999999999992e-24 < b

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

      2. associate-*l/ 74.0%

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

      3. associate-*r/ 74.0%

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

      4. associate-/l* 73.9%

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

      5. sub-neg 73.9%

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

      6. distribute-neg-frac 73.9%

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

      7. metadata-eval 73.9%

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

      8. associate-*r/ 74.0%

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

      9. *-rgt-identity 74.0%

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

      10. difference-of-squares 87.8%

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

      11. associate-/r* 87.8%

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

   3. Simplified 87.8%

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

      1. clear-num 87.5%

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

      2. inv-pow 87.5%

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

      3. associate-/r/ 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. unpow-1 87.6%

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

      2. associate-*l/ 87.6%

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

      3. +-commutative 87.6%

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

      4. metadata-eval 87.6%

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

      5. distribute-neg-frac 87.6%

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

      6. sub-neg 87.6%

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

   7. Simplified 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. associate-/l/ 98.7%

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

      3. inv-pow 98.7%

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

      4. inv-pow 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. expm1-log1p-u 85.9%

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

       2. expm1-udef 58.3%

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

       3. *-un-lft-identity 58.3%

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

       4. inv-pow 58.3%

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

       5. unpow-1 58.3%

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

       6. +-commutative 58.3%

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

   11. Applied egg-rr 58.3%

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

       1. expm1-def 85.9%

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

       2. expm1-log1p 98.7%

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

       3. associate-/r/ 99.6%

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

       4. +-commutative 99.6%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in a around 0 95.1%

       \[\leadsto \frac{1}{2 \cdot \left(b - a\right)} \cdot \color{blue}{\frac{\pi}{a \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b - a\right)} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]

Alternative 6: 75.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.4e-24) (* 0.5 (/ PI (* a (* b a)))) (* 0.5 (/ PI (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.4e-24) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.4e-24) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else {
		tmp = 0.5 * (Math.PI / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.4e-24:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	else:
		tmp = 0.5 * (math.pi / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.4e-24)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.4e-24)
		tmp = 0.5 * (pi / (a * (b * a)));
	else
		tmp = 0.5 * (pi / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.4e-24], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.39999999999999992e-24

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l/ 79.9%

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

      3. associate-*r/ 79.9%

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

      4. associate-/l* 79.9%

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

      5. sub-neg 79.9%

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

      6. distribute-neg-frac 79.9%

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

      7. metadata-eval 79.9%

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

      8. associate-*r/ 79.8%

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

      9. *-rgt-identity 79.8%

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

      10. difference-of-squares 87.2%

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

      11. associate-/r* 87.1%

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

   3. Simplified 87.1%

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

      1. clear-num 87.1%

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

      2. inv-pow 87.1%

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

      3. associate-/r/ 87.1%

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

   5. Applied egg-rr 87.1%

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

      1. unpow-1 87.1%

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

      2. associate-*l/ 87.2%

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

      3. +-commutative 87.2%

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

      4. metadata-eval 87.2%

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

      5. distribute-neg-frac 87.2%

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

      6. sub-neg 87.2%

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

   7. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. associate-/l/ 99.4%

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

      3. inv-pow 99.4%

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

      4. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in b around 0 59.5%

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

       1. *-commutative 59.5%

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

       2. *-commutative 59.5%

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

       3. unpow2 59.5%

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

       4. associate-*r* 69.3%

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

   12. Simplified 69.3%

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

   if 3.39999999999999992e-24 < b

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

      2. associate-*l/ 74.0%

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

      3. associate-*r/ 74.0%

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

      4. associate-/l* 73.9%

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

      5. sub-neg 73.9%

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

      6. distribute-neg-frac 73.9%

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

      7. metadata-eval 73.9%

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

      8. associate-*r/ 74.0%

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

      9. *-rgt-identity 74.0%

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

      10. difference-of-squares 87.8%

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

      11. associate-/r* 87.8%

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

   3. Simplified 87.8%

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

      1. clear-num 87.5%

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

      2. inv-pow 87.5%

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

      3. associate-/r/ 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. unpow-1 87.6%

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

      2. associate-*l/ 87.6%

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

      3. +-commutative 87.6%

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

      4. metadata-eval 87.6%

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

      5. distribute-neg-frac 87.6%

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

      6. sub-neg 87.6%

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

   7. Simplified 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. associate-/l/ 98.7%

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

      3. inv-pow 98.7%

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

      4. inv-pow 98.7%

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

   9. Applied egg-rr 98.7%

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

   10. Taylor expanded in b around inf 80.8%

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

       1. *-commutative 80.8%

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

       2. unpow2 80.8%

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

       3. associate-*l* 92.0%

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

       4. *-commutative 92.0%

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

   12. Simplified 92.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 7: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.1e-24) (* 0.5 (/ (/ PI a) (* b a))) (* 0.5 (/ PI (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.1e-24) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * a));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.1e-24) {
		tmp = 0.5 * ((Math.PI / a) / (b * a));
	} else {
		tmp = 0.5 * (Math.PI / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.1e-24:
		tmp = 0.5 * ((math.pi / a) / (b * a))
	else:
		tmp = 0.5 * (math.pi / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.1e-24)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * a)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.1e-24)
		tmp = 0.5 * ((pi / a) / (b * a));
	else
		tmp = 0.5 * (pi / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.1e-24], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.1e-24

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l/ 79.9%

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

      3. associate-*r/ 79.9%

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

      4. associate-/l* 79.9%

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

      5. sub-neg 79.9%

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

      6. distribute-neg-frac 79.9%

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

      7. metadata-eval 79.9%

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

      8. associate-*r/ 79.8%

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

      9. *-rgt-identity 79.8%

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

      10. difference-of-squares 87.2%

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

      11. associate-/r* 87.1%

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

   3. Simplified 87.1%

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

      1. clear-num 87.1%

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

      2. inv-pow 87.1%

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

      3. associate-/r/ 87.1%

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

   5. Applied egg-rr 87.1%

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

      1. unpow-1 87.1%

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

      2. associate-*l/ 87.2%

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

      3. +-commutative 87.2%

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

      4. metadata-eval 87.2%

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

      5. distribute-neg-frac 87.2%

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

      6. sub-neg 87.2%

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

   7. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. associate-/l/ 99.4%

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

      3. inv-pow 99.4%

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

      4. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in b around 0 59.5%

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

       1. *-commutative 59.5%

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

       2. unpow2 59.5%

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

       3. associate-*r* 69.3%

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

       4. *-commutative 69.3%

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

       5. associate-/r* 69.1%

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

       6. *-commutative 69.1%

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

   12. Simplified 69.1%

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

   if 3.1e-24 < b

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

      2. associate-*l/ 74.0%

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

      3. associate-*r/ 74.0%

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

      4. associate-/l* 73.9%

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

      5. sub-neg 73.9%

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

      6. distribute-neg-frac 73.9%

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

      7. metadata-eval 73.9%

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

      8. associate-*r/ 74.0%

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

      9. *-rgt-identity 74.0%

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

      10. difference-of-squares 87.8%

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

      11. associate-/r* 87.8%

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

   3. Simplified 87.8%

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

      1. clear-num 87.5%

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

      2. inv-pow 87.5%

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

      3. associate-/r/ 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. unpow-1 87.6%

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

      2. associate-*l/ 87.6%

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

      3. +-commutative 87.6%

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

      4. metadata-eval 87.6%

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

      5. distribute-neg-frac 87.6%

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

      6. sub-neg 87.6%

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

   7. Simplified 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. associate-/l/ 98.7%

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

      3. inv-pow 98.7%

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

      4. inv-pow 98.7%

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

   9. Applied egg-rr 98.7%

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

   10. Taylor expanded in b around inf 80.8%

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

       1. *-commutative 80.8%

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

       2. unpow2 80.8%

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

       3. associate-*l* 92.0%

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

       4. *-commutative 92.0%

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

   12. Simplified 92.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 8: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{b \cdot \left(b \cdot a\right)}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-24) (* 0.5 (/ (/ PI a) (* b a))) (/ 0.5 (/ (* b (* b a)) PI))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-24) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * a));
	} else {
		tmp = 0.5 / ((b * (b * a)) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-24) {
		tmp = 0.5 * ((Math.PI / a) / (b * a));
	} else {
		tmp = 0.5 / ((b * (b * a)) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.5e-24:
		tmp = 0.5 * ((math.pi / a) / (b * a))
	else:
		tmp = 0.5 / ((b * (b * a)) / math.pi)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e-24)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * a)));
	else
		tmp = Float64(0.5 / Float64(Float64(b * Float64(b * a)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.5e-24)
		tmp = 0.5 * ((pi / a) / (b * a));
	else
		tmp = 0.5 / ((b * (b * a)) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e-24], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.4999999999999996e-24

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l/ 79.9%

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

      3. associate-*r/ 79.9%

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

      4. associate-/l* 79.9%

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

      5. sub-neg 79.9%

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

      6. distribute-neg-frac 79.9%

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

      7. metadata-eval 79.9%

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

      8. associate-*r/ 79.8%

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

      9. *-rgt-identity 79.8%

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

      10. difference-of-squares 87.2%

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

      11. associate-/r* 87.1%

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

   3. Simplified 87.1%

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

      1. clear-num 87.1%

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

      2. inv-pow 87.1%

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

      3. associate-/r/ 87.1%

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

   5. Applied egg-rr 87.1%

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

      1. unpow-1 87.1%

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

      2. associate-*l/ 87.2%

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

      3. +-commutative 87.2%

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

      4. metadata-eval 87.2%

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

      5. distribute-neg-frac 87.2%

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

      6. sub-neg 87.2%

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

   7. Simplified 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. associate-/l/ 99.4%

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

      3. inv-pow 99.4%

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

      4. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in b around 0 59.5%

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

       1. *-commutative 59.5%

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

       2. unpow2 59.5%

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

       3. associate-*r* 69.3%

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

       4. *-commutative 69.3%

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

       5. associate-/r* 69.1%

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

       6. *-commutative 69.1%

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

   12. Simplified 69.1%

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

   if 3.4999999999999996e-24 < b

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

      2. associate-*l/ 74.0%

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

      3. associate-*r/ 74.0%

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

      4. associate-/l* 73.9%

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

      5. sub-neg 73.9%

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

      6. distribute-neg-frac 73.9%

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

      7. metadata-eval 73.9%

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

      8. associate-*r/ 74.0%

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

      9. *-rgt-identity 74.0%

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

      10. difference-of-squares 87.8%

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

      11. associate-/r* 87.8%

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

   3. Simplified 87.8%

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

      1. clear-num 87.5%

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

      2. inv-pow 87.5%

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

      3. associate-/r/ 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. unpow-1 87.6%

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

      2. associate-*l/ 87.6%

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

      3. +-commutative 87.6%

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

      4. metadata-eval 87.6%

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

      5. distribute-neg-frac 87.6%

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

      6. sub-neg 87.6%

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

   7. Simplified 87.6%

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

      1. *-un-lft-identity 87.6%

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

      2. associate-/l/ 98.7%

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

      3. inv-pow 98.7%

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

      4. inv-pow 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. expm1-log1p-u 85.9%

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

       2. expm1-udef 58.3%

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

       3. *-un-lft-identity 58.3%

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

       4. inv-pow 58.3%

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

       5. unpow-1 58.3%

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

       6. +-commutative 58.3%

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

   11. Applied egg-rr 58.3%

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

       1. expm1-def 85.9%

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

       2. expm1-log1p 98.7%

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

       3. associate-/r/ 99.6%

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

       4. +-commutative 99.6%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in b around inf 80.8%

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

       1. associate-*r/ 80.8%

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

       2. associate-/l* 80.8%

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

       3. *-commutative 80.8%

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

       4. unpow2 80.8%

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

       5. associate-*l* 92.0%

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

       6. *-commutative 92.0%

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

   16. Simplified 92.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{b \cdot \left(b \cdot a\right)}{\pi}}\\ \end{array} \]

Alternative 9: 63.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-*l/ 78.4%

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

   3. associate-*r/ 78.4%

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

   4. associate-/l* 78.4%

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

   5. sub-neg 78.4%

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

   6. distribute-neg-frac 78.4%

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

   7. metadata-eval 78.4%

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

   8. associate-*r/ 78.3%

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

   9. *-rgt-identity 78.3%

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

   10. difference-of-squares 87.3%

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

   11. associate-/r* 87.3%

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

3. Simplified 87.3%

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

   1. clear-num 87.2%

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

   2. inv-pow 87.2%

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

   3. associate-/r/ 87.2%

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

5. Applied egg-rr 87.2%

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

   1. unpow-1 87.2%

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

   2. associate-*l/ 87.3%

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

   3. +-commutative 87.3%

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

   4. metadata-eval 87.3%

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

   5. distribute-neg-frac 87.3%

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

   6. sub-neg 87.3%

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

7. Simplified 87.3%

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

   1. *-un-lft-identity 87.3%

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

   2. associate-/l/ 99.2%

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

   3. inv-pow 99.2%

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

   4. inv-pow 99.2%

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

9. Applied egg-rr 99.2%

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

10. Taylor expanded in b around 0 53.7%

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

    1. *-commutative 53.7%

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

    2. *-commutative 53.7%

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

    3. unpow2 53.7%

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

    4. associate-*r* 61.0%

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

12. Simplified 61.0%

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

13. Final simplification 61.0%

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

Reproduce

herbie shell --seed 2023279 
(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))))