NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 9.4s

Alternatives: 11

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: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. times-frac 76.7%

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

   2. *-commutative 76.7%

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

   3. times-frac 76.7%

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

   4. difference-of-squares 84.1%

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

   5. associate-/r* 85.4%

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

   6. metadata-eval 85.4%

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

   7. sub-neg 85.4%

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

   8. distribute-neg-frac 85.4%

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

   9. metadata-eval 85.4%

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

3. Simplified 85.4%

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

   1. div-inv 85.4%

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

5. Applied egg-rr 85.4%

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

   1. distribute-lft-in 82.2%

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

   2. associate-*l/ 82.2%

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

   3. un-div-inv 82.2%

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

   4. associate-*l/ 82.2%

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

   5. un-div-inv 82.2%

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

7. Applied egg-rr 82.2%

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

   1. metadata-eval 82.2%

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

   2. distribute-neg-frac 82.2%

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

   3. distribute-lft-out 85.4%

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

   4. associate-*l/ 85.4%

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

   5. associate-/r* 84.1%

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

   6. sub-neg 84.1%

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

   7. *-commutative 84.1%

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

   8. sub-neg 84.1%

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

   9. distribute-neg-frac 84.1%

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

   10. metadata-eval 84.1%

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

   11. *-commutative 84.1%

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

   12. times-frac 85.4%

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

   13. +-commutative 85.4%

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

9. Simplified 85.4%

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

    1. associate-*l/ 85.4%

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

11. Applied egg-rr 85.4%

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

    1. associate-*r/ 99.7%

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

13. Applied egg-rr 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-226}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e-226)
   (* (/ -1.0 b) (* 0.5 (/ (/ PI (+ a b)) (- b a))))
   (if (<= b 1.6e+104)
     (* (/ PI (- (* b b) (* a a))) (+ (/ 0.5 a) (/ -0.5 b)))
     (/ (/ 0.5 b) (/ (* a b) PI)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e-226) {
		tmp = (-1.0 / b) * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	} else if (b <= 1.6e+104) {
		tmp = (((double) M_PI) / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = (0.5 / b) / ((a * b) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2e-226) {
		tmp = (-1.0 / b) * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	} else if (b <= 1.6e+104) {
		tmp = (Math.PI / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = (0.5 / b) / ((a * b) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2e-226:
		tmp = (-1.0 / b) * (0.5 * ((math.pi / (a + b)) / (b - a)))
	elif b <= 1.6e+104:
		tmp = (math.pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b))
	else:
		tmp = (0.5 / b) / ((a * b) / math.pi)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e-226)
		tmp = Float64(Float64(-1.0 / b) * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	elseif (b <= 1.6e+104)
		tmp = Float64(Float64(pi / Float64(Float64(b * b) - Float64(a * a))) * Float64(Float64(0.5 / a) + Float64(-0.5 / b)));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(Float64(a * b) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2e-226)
		tmp = (-1.0 / b) * (0.5 * ((pi / (a + b)) / (b - a)));
	elseif (b <= 1.6e+104)
		tmp = (pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	else
		tmp = (0.5 / b) / ((a * b) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e-226], N[(N[(-1.0 / b), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+104], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.99999999999999984e-226

   1. Initial program 79.1%

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

      1. times-frac 79.1%

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

      2. *-commutative 79.1%

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

      3. times-frac 79.1%

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

      4. difference-of-squares 86.4%

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

      5. associate-/r* 87.1%

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

      6. metadata-eval 87.1%

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

      7. sub-neg 87.1%

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

      8. distribute-neg-frac 87.1%

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

      9. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in a around inf 67.3%

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

   if 1.99999999999999984e-226 < b < 1.6e104

   1. Initial program 88.6%

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

      1. times-frac 88.8%

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

      2. *-commutative 88.8%

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

      3. times-frac 88.8%

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

      4. difference-of-squares 88.8%

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

      5. associate-/r* 89.9%

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

      6. metadata-eval 89.9%

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

      7. sub-neg 89.9%

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

      8. distribute-neg-frac 89.9%

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

      9. metadata-eval 89.9%

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

   3. Simplified 89.9%

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

      1. distribute-lft-in 84.9%

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

      2. associate-/l/ 84.9%

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

      3. associate-/l/ 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. distribute-lft-out 88.8%

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

      2. associate-*r* 88.8%

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

      3. associate-*l/ 88.6%

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

      4. *-commutative 88.6%

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

      5. difference-of-squares 88.6%

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

      6. associate-*l/ 88.8%

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

      7. distribute-lft-in 88.8%

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

      8. associate-*r/ 88.8%

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

      9. metadata-eval 88.8%

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

      10. associate-*r/ 88.8%

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

      11. metadata-eval 88.8%

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

   7. Simplified 88.8%

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

   if 1.6e104 < b

   1. Initial program 53.0%

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

   2. Taylor expanded in b around inf 70.4%

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

      1. unpow2 70.4%

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

      2. associate-/r* 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in a around 0 73.8%

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

      1. un-div-inv 73.9%

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

      2. div-inv 73.9%

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

      3. metadata-eval 73.9%

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

      4. associate-*r/ 73.9%

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

      5. *-commutative 73.9%

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

      6. div-inv 73.9%

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

      7. associate-/l* 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.8%

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

      2. expm1-udef 66.2%

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

      3. associate-/l/ 66.2%

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

      4. associate-/r/ 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. expm1-def 99.7%

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

       2. expm1-log1p 99.9%

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

       3. associate-/l* 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-226}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \]

Alternative 3: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+104}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e+104)
   (* (+ (/ 1.0 a) (/ -1.0 b)) (* 0.5 (/ (/ PI (+ a b)) (- b a))))
   (/ (/ 0.5 b) (/ (* a b) PI))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+104) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	} else {
		tmp = (0.5 / b) / ((a * b) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+104) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	} else {
		tmp = (0.5 / b) / ((a * b) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.5e+104:
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((math.pi / (a + b)) / (b - a)))
	else:
		tmp = (0.5 / b) / ((a * b) / math.pi)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e+104)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(Float64(a * b) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.5e+104)
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((pi / (a + b)) / (b - a)));
	else
		tmp = (0.5 / b) / ((a * b) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e+104], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999984e104

   1. Initial program 81.8%

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

      1. times-frac 81.9%

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

      2. *-commutative 81.9%

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

      3. times-frac 81.9%

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

      4. difference-of-squares 87.1%

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

      5. associate-/r* 87.9%

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

      6. metadata-eval 87.9%

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

      7. sub-neg 87.9%

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

      8. distribute-neg-frac 87.9%

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

      9. metadata-eval 87.9%

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

   3. Simplified 87.9%

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

   if 1.49999999999999984e104 < b

   1. Initial program 53.0%

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

   2. Taylor expanded in b around inf 70.4%

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

      1. unpow2 70.4%

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

      2. associate-/r* 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in a around 0 73.8%

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

      1. un-div-inv 73.9%

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

      2. div-inv 73.9%

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

      3. metadata-eval 73.9%

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

      4. associate-*r/ 73.9%

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

      5. *-commutative 73.9%

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

      6. div-inv 73.9%

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

      7. associate-/l* 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.8%

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

      2. expm1-udef 66.2%

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

      3. associate-/l/ 66.2%

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

      4. associate-/r/ 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. expm1-def 99.7%

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

       2. expm1-log1p 99.9%

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

       3. associate-/l* 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+104}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \]

Alternative 4: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+103}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\pi \cdot \frac{0.5}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e+103)
   (* (+ (/ 1.0 a) (/ -1.0 b)) (/ (* PI (/ 0.5 (+ a b))) (- b a)))
   (/ (/ 0.5 b) (/ (* a b) PI))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e+103) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((((double) M_PI) * (0.5 / (a + b))) / (b - a));
	} else {
		tmp = (0.5 / b) / ((a * b) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1e+103) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((Math.PI * (0.5 / (a + b))) / (b - a));
	} else {
		tmp = (0.5 / b) / ((a * b) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e+103:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((math.pi * (0.5 / (a + b))) / (b - a))
	else:
		tmp = (0.5 / b) / ((a * b) / math.pi)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e+103)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(pi * Float64(0.5 / Float64(a + b))) / Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(Float64(a * b) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e+103)
		tmp = ((1.0 / a) + (-1.0 / b)) * ((pi * (0.5 / (a + b))) / (b - a));
	else
		tmp = (0.5 / b) / ((a * b) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e+103], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1e103

   1. Initial program 81.8%

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

      1. times-frac 81.9%

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

      2. *-commutative 81.9%

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

      3. times-frac 81.9%

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

      4. difference-of-squares 87.1%

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

      5. associate-/r* 87.9%

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

      6. metadata-eval 87.9%

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

      7. sub-neg 87.9%

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

      8. distribute-neg-frac 87.9%

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

      9. metadata-eval 87.9%

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

   3. Simplified 87.9%

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

      1. div-inv 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. distribute-lft-in 84.1%

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

      2. associate-*l/ 84.1%

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

      3. un-div-inv 84.1%

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

      4. associate-*l/ 84.1%

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

      5. un-div-inv 84.1%

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

   7. Applied egg-rr 84.1%

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

      1. metadata-eval 84.1%

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

      2. distribute-neg-frac 84.1%

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

      3. distribute-lft-out 87.9%

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

      4. associate-*l/ 87.9%

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

      5. associate-/r* 87.1%

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

      6. sub-neg 87.1%

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

      7. *-commutative 87.1%

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

      8. sub-neg 87.1%

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

      9. distribute-neg-frac 87.1%

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

      10. metadata-eval 87.1%

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

      11. *-commutative 87.1%

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

      12. times-frac 87.9%

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

      13. +-commutative 87.9%

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

   9. Simplified 87.9%

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

       1. associate-*l/ 87.9%

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

   11. Applied egg-rr 87.9%

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

   if 1e103 < b

   1. Initial program 53.0%

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

   2. Taylor expanded in b around inf 70.4%

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

      1. unpow2 70.4%

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

      2. associate-/r* 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in a around 0 73.8%

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

      1. un-div-inv 73.9%

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

      2. div-inv 73.9%

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

      3. metadata-eval 73.9%

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

      4. associate-*r/ 73.9%

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

      5. *-commutative 73.9%

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

      6. div-inv 73.9%

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

      7. associate-/l* 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.8%

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

      2. expm1-udef 66.2%

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

      3. associate-/l/ 66.2%

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

      4. associate-/r/ 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. expm1-def 99.7%

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

       2. expm1-log1p 99.9%

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

       3. associate-/l* 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+103}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\pi \cdot \frac{0.5}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \]

Alternative 5: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\\ \mathbf{if}\;b \leq 2.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-1}{b} \cdot t_0\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (/ PI (+ a b)) (- b a)))))
   (if (<= b 2.1e-102)
     (* (/ -1.0 b) t_0)
     (if (<= b 1e+103) (* (/ 1.0 a) t_0) (/ (/ 0.5 b) (/ (* a b) PI))))))

C

double code(double a, double b) {
	double t_0 = 0.5 * ((((double) M_PI) / (a + b)) / (b - a));
	double tmp;
	if (b <= 2.1e-102) {
		tmp = (-1.0 / b) * t_0;
	} else if (b <= 1e+103) {
		tmp = (1.0 / a) * t_0;
	} else {
		tmp = (0.5 / b) / ((a * b) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 * ((Math.PI / (a + b)) / (b - a));
	double tmp;
	if (b <= 2.1e-102) {
		tmp = (-1.0 / b) * t_0;
	} else if (b <= 1e+103) {
		tmp = (1.0 / a) * t_0;
	} else {
		tmp = (0.5 / b) / ((a * b) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 * ((math.pi / (a + b)) / (b - a))
	tmp = 0
	if b <= 2.1e-102:
		tmp = (-1.0 / b) * t_0
	elif b <= 1e+103:
		tmp = (1.0 / a) * t_0
	else:
		tmp = (0.5 / b) / ((a * b) / math.pi)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a)))
	tmp = 0.0
	if (b <= 2.1e-102)
		tmp = Float64(Float64(-1.0 / b) * t_0);
	elseif (b <= 1e+103)
		tmp = Float64(Float64(1.0 / a) * t_0);
	else
		tmp = Float64(Float64(0.5 / b) / Float64(Float64(a * b) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 * ((pi / (a + b)) / (b - a));
	tmp = 0.0;
	if (b <= 2.1e-102)
		tmp = (-1.0 / b) * t_0;
	elseif (b <= 1e+103)
		tmp = (1.0 / a) * t_0;
	else
		tmp = (0.5 / b) / ((a * b) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.1e-102], N[(N[(-1.0 / b), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[b, 1e+103], N[(N[(1.0 / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.1e-102

   1. Initial program 78.9%

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

      1. times-frac 79.0%

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

      2. *-commutative 79.0%

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

      3. times-frac 79.0%

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

      4. difference-of-squares 85.2%

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

      5. associate-/r* 85.8%

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

      6. metadata-eval 85.8%

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

      7. sub-neg 85.8%

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

      8. distribute-neg-frac 85.8%

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

      9. metadata-eval 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in a around inf 67.7%

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

   if 2.1e-102 < b < 1e103

   1. Initial program 96.7%

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

      1. times-frac 96.9%

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

      2. *-commutative 96.9%

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

      3. times-frac 96.9%

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

      4. difference-of-squares 96.9%

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

      5. associate-/r* 98.9%

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

      6. metadata-eval 98.9%

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

      7. sub-neg 98.9%

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

      8. distribute-neg-frac 98.9%

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

      9. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in a around 0 70.6%

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

   if 1e103 < b

   1. Initial program 53.0%

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

   2. Taylor expanded in b around inf 70.4%

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

      1. unpow2 70.4%

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

      2. associate-/r* 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in a around 0 73.8%

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

      1. un-div-inv 73.9%

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

      2. div-inv 73.9%

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

      3. metadata-eval 73.9%

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

      4. associate-*r/ 73.9%

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

      5. *-commutative 73.9%

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

      6. div-inv 73.9%

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

      7. associate-/l* 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.8%

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

      2. expm1-udef 66.2%

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

      3. associate-/l/ 66.2%

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

      4. associate-/r/ 66.2%

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

   9. Applied egg-rr 66.2%

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

       1. expm1-def 99.7%

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

       2. expm1-log1p 99.9%

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

       3. associate-/l* 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \]

Alternative 6: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e-11)
   (* 0.5 (/ (/ PI (* a a)) b))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-11) {
		tmp = 0.5 * ((((double) M_PI) / (a * a)) / b);
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-11) {
		tmp = 0.5 * ((Math.PI / (a * a)) / b);
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.65e-11:
		tmp = 0.5 * ((math.pi / (a * a)) / b)
	else:
		tmp = (math.pi / a) * (0.5 / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e-11)
		tmp = Float64(0.5 * Float64(Float64(pi / Float64(a * a)) / b));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.65e-11)
		tmp = 0.5 * ((pi / (a * a)) / b);
	else
		tmp = (pi / a) * (0.5 / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e-11], N[(0.5 * N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6500000000000001e-11

   1. Initial program 80.0%

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

      1. times-frac 80.1%

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

      2. *-commutative 80.1%

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

      3. times-frac 80.1%

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

      4. difference-of-squares 85.9%

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

      5. associate-/r* 86.7%

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

      6. metadata-eval 86.7%

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

      7. sub-neg 86.7%

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

      8. distribute-neg-frac 86.7%

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

      9. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. div-inv 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. distribute-lft-in 82.5%

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

      2. associate-*l/ 82.5%

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

      3. un-div-inv 82.5%

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

      4. associate-*l/ 82.5%

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

      5. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. metadata-eval 82.5%

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

      2. distribute-neg-frac 82.5%

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

      3. distribute-lft-out 86.7%

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

      4. associate-*l/ 86.7%

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

      5. associate-/r* 85.9%

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

      6. sub-neg 85.9%

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

      7. *-commutative 85.9%

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

      8. sub-neg 85.9%

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

      9. distribute-neg-frac 85.9%

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

      10. metadata-eval 85.9%

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

      11. *-commutative 85.9%

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

      12. times-frac 86.7%

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

      13. +-commutative 86.7%

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

   9. Simplified 86.7%

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

       1. associate-*l/ 86.7%

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

   11. Applied egg-rr 86.7%

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

   12. Taylor expanded in a around inf 57.3%

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

       1. associate-/r* 57.3%

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

       2. unpow2 57.3%

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

   14. Simplified 57.3%

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

   if 1.6500000000000001e-11 < b

   1. Initial program 66.5%

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

      1. times-frac 66.7%

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

      2. *-commutative 66.7%

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

      3. times-frac 66.7%

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

      4. difference-of-squares 79.0%

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

      5. associate-/r* 81.4%

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

      6. metadata-eval 81.4%

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

      7. sub-neg 81.4%

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

      8. distribute-neg-frac 81.4%

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

      9. metadata-eval 81.4%

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

   3. Simplified 81.4%

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

      1. div-inv 81.4%

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

   5. Applied egg-rr 81.4%

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

   6. Taylor expanded in b around inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. *-commutative 70.2%

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

      3. times-frac 70.0%

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

      4. unpow2 70.0%

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

   8. Simplified 70.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 7: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.75e-11)
   (* (/ PI b) (/ 0.5 (* a a)))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.75e-11) {
		tmp = (((double) M_PI) / b) * (0.5 / (a * a));
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.75e-11) {
		tmp = (Math.PI / b) * (0.5 / (a * a));
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.75e-11:
		tmp = (math.pi / b) * (0.5 / (a * a))
	else:
		tmp = (math.pi / a) * (0.5 / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.75e-11)
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * a)));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.75e-11)
		tmp = (pi / b) * (0.5 / (a * a));
	else
		tmp = (pi / a) * (0.5 / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.75e-11], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7500000000000001e-11

   1. Initial program 80.0%

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

      1. times-frac 80.1%

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

      2. *-commutative 80.1%

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

      3. times-frac 80.1%

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

      4. difference-of-squares 85.9%

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

      5. associate-/r* 86.7%

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

      6. metadata-eval 86.7%

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

      7. sub-neg 86.7%

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

      8. distribute-neg-frac 86.7%

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

      9. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. div-inv 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. distribute-lft-in 82.5%

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

      2. associate-*l/ 82.5%

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

      3. un-div-inv 82.5%

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

      4. associate-*l/ 82.5%

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

      5. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. metadata-eval 82.5%

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

      2. distribute-neg-frac 82.5%

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

      3. distribute-lft-out 86.7%

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

      4. associate-*l/ 86.7%

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

      5. associate-/r* 85.9%

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

      6. sub-neg 85.9%

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

      7. *-commutative 85.9%

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

      8. sub-neg 85.9%

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

      9. distribute-neg-frac 85.9%

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

      10. metadata-eval 85.9%

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

      11. *-commutative 85.9%

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

      12. times-frac 86.7%

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

      13. +-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in a around inf 57.3%

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

       1. associate-*r/ 57.3%

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

       2. *-commutative 57.3%

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

       3. *-commutative 57.3%

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

       4. times-frac 57.2%

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

       5. unpow2 57.2%

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

   12. Simplified 57.2%

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

   if 1.7500000000000001e-11 < b

   1. Initial program 66.5%

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

      1. times-frac 66.7%

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

      2. *-commutative 66.7%

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

      3. times-frac 66.7%

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

      4. difference-of-squares 79.0%

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

      5. associate-/r* 81.4%

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

      6. metadata-eval 81.4%

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

      7. sub-neg 81.4%

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

      8. distribute-neg-frac 81.4%

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

      9. metadata-eval 81.4%

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

   3. Simplified 81.4%

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

      1. div-inv 81.4%

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

   5. Applied egg-rr 81.4%

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

   6. Taylor expanded in b around inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. *-commutative 70.2%

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

      3. times-frac 70.0%

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

      4. unpow2 70.0%

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

   8. Simplified 70.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 8: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.36e-11)
   (* (/ PI b) (/ 0.5 (* a a)))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.36e-11) {
		tmp = (((double) M_PI) / b) * (0.5 / (a * a));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.36e-11) {
		tmp = (Math.PI / b) * (0.5 / (a * a));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.36e-11:
		tmp = (math.pi / b) * (0.5 / (a * a))
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.36e-11)
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * a)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.36e-11)
		tmp = (pi / b) * (0.5 / (a * a));
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.36e-11], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.36e-11

   1. Initial program 80.0%

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

      1. times-frac 80.1%

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

      2. *-commutative 80.1%

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

      3. times-frac 80.1%

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

      4. difference-of-squares 85.9%

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

      5. associate-/r* 86.7%

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

      6. metadata-eval 86.7%

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

      7. sub-neg 86.7%

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

      8. distribute-neg-frac 86.7%

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

      9. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. div-inv 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. distribute-lft-in 82.5%

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

      2. associate-*l/ 82.5%

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

      3. un-div-inv 82.5%

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

      4. associate-*l/ 82.5%

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

      5. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. metadata-eval 82.5%

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

      2. distribute-neg-frac 82.5%

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

      3. distribute-lft-out 86.7%

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

      4. associate-*l/ 86.7%

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

      5. associate-/r* 85.9%

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

      6. sub-neg 85.9%

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

      7. *-commutative 85.9%

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

      8. sub-neg 85.9%

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

      9. distribute-neg-frac 85.9%

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

      10. metadata-eval 85.9%

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

      11. *-commutative 85.9%

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

      12. times-frac 86.7%

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

      13. +-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in a around inf 57.3%

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

       1. associate-*r/ 57.3%

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

       2. *-commutative 57.3%

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

       3. *-commutative 57.3%

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

       4. times-frac 57.2%

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

       5. unpow2 57.2%

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

   12. Simplified 57.2%

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

   if 1.36e-11 < b

   1. Initial program 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-/r/ 66.6%

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

      3. associate-*l/ 66.6%

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

      4. *-commutative 66.6%

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

      5. associate-/r/ 66.6%

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

      6. times-frac 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in b around inf 70.2%

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

      1. unpow2 70.2%

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

   6. Simplified 70.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 9: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{b}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.05e-11)
   (* (/ PI b) (/ 0.5 (* a a)))
   (* (/ PI b) (/ (/ 0.5 b) a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.05e-11) {
		tmp = (((double) M_PI) / b) * (0.5 / (a * a));
	} else {
		tmp = (((double) M_PI) / b) * ((0.5 / b) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.05e-11) {
		tmp = (Math.PI / b) * (0.5 / (a * a));
	} else {
		tmp = (Math.PI / b) * ((0.5 / b) / a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.05e-11:
		tmp = (math.pi / b) * (0.5 / (a * a))
	else:
		tmp = (math.pi / b) * ((0.5 / b) / a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.05e-11)
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * a)));
	else
		tmp = Float64(Float64(pi / b) * Float64(Float64(0.5 / b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.05e-11)
		tmp = (pi / b) * (0.5 / (a * a));
	else
		tmp = (pi / b) * ((0.5 / b) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.05e-11], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0499999999999999e-11

   1. Initial program 80.0%

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

      1. times-frac 80.1%

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

      2. *-commutative 80.1%

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

      3. times-frac 80.1%

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

      4. difference-of-squares 85.9%

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

      5. associate-/r* 86.7%

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

      6. metadata-eval 86.7%

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

      7. sub-neg 86.7%

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

      8. distribute-neg-frac 86.7%

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

      9. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. div-inv 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. distribute-lft-in 82.5%

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

      2. associate-*l/ 82.5%

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

      3. un-div-inv 82.5%

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

      4. associate-*l/ 82.5%

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

      5. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. metadata-eval 82.5%

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

      2. distribute-neg-frac 82.5%

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

      3. distribute-lft-out 86.7%

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

      4. associate-*l/ 86.7%

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

      5. associate-/r* 85.9%

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

      6. sub-neg 85.9%

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

      7. *-commutative 85.9%

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

      8. sub-neg 85.9%

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

      9. distribute-neg-frac 85.9%

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

      10. metadata-eval 85.9%

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

      11. *-commutative 85.9%

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

      12. times-frac 86.7%

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

      13. +-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in a around inf 57.3%

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

       1. associate-*r/ 57.3%

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

       2. *-commutative 57.3%

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

       3. *-commutative 57.3%

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

       4. times-frac 57.2%

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

       5. unpow2 57.2%

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

   12. Simplified 57.2%

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

   if 1.0499999999999999e-11 < b

   1. Initial program 66.5%

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

   2. Taylor expanded in b around inf 62.7%

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

      1. unpow2 62.7%

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

      2. associate-/r* 65.1%

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

   4. Simplified 65.1%

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

   5. Taylor expanded in a around 0 72.4%

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

      1. un-div-inv 72.4%

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

      2. div-inv 72.4%

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

      3. metadata-eval 72.4%

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

      4. associate-*r/ 72.5%

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

      5. *-commutative 72.5%

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

      6. div-inv 72.5%

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

      7. associate-/l* 72.5%

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

   7. Applied egg-rr 72.5%

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

      1. expm1-log1p-u 68.8%

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

      2. expm1-udef 62.3%

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

      3. associate-/l/ 62.3%

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

      4. associate-/r/ 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. expm1-def 87.2%

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

       2. expm1-log1p 90.8%

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

       3. times-frac 90.7%

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

   11. Simplified 90.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{b}}{a}\\ \end{array} \]

Alternative 10: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.75e-11)
   (* (/ PI b) (/ 0.5 (* a a)))
   (/ (/ 0.5 b) (/ (* a b) PI))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.75e-11) {
		tmp = (((double) M_PI) / b) * (0.5 / (a * a));
	} else {
		tmp = (0.5 / b) / ((a * b) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.75e-11) {
		tmp = (Math.PI / b) * (0.5 / (a * a));
	} else {
		tmp = (0.5 / b) / ((a * b) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.75e-11:
		tmp = (math.pi / b) * (0.5 / (a * a))
	else:
		tmp = (0.5 / b) / ((a * b) / math.pi)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.75e-11)
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * a)));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(Float64(a * b) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.75e-11)
		tmp = (pi / b) * (0.5 / (a * a));
	else
		tmp = (0.5 / b) / ((a * b) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.75e-11], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7500000000000001e-11

   1. Initial program 80.0%

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

      1. times-frac 80.1%

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

      2. *-commutative 80.1%

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

      3. times-frac 80.1%

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

      4. difference-of-squares 85.9%

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

      5. associate-/r* 86.7%

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

      6. metadata-eval 86.7%

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

      7. sub-neg 86.7%

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

      8. distribute-neg-frac 86.7%

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

      9. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. div-inv 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. distribute-lft-in 82.5%

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

      2. associate-*l/ 82.5%

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

      3. un-div-inv 82.5%

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

      4. associate-*l/ 82.5%

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

      5. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. metadata-eval 82.5%

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

      2. distribute-neg-frac 82.5%

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

      3. distribute-lft-out 86.7%

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

      4. associate-*l/ 86.7%

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

      5. associate-/r* 85.9%

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

      6. sub-neg 85.9%

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

      7. *-commutative 85.9%

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

      8. sub-neg 85.9%

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

      9. distribute-neg-frac 85.9%

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

      10. metadata-eval 85.9%

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

      11. *-commutative 85.9%

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

      12. times-frac 86.7%

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

      13. +-commutative 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in a around inf 57.3%

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

       1. associate-*r/ 57.3%

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

       2. *-commutative 57.3%

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

       3. *-commutative 57.3%

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

       4. times-frac 57.2%

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

       5. unpow2 57.2%

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

   12. Simplified 57.2%

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

   if 1.7500000000000001e-11 < b

   1. Initial program 66.5%

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

   2. Taylor expanded in b around inf 62.7%

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

      1. unpow2 62.7%

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

      2. associate-/r* 65.1%

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

   4. Simplified 65.1%

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

   5. Taylor expanded in a around 0 72.4%

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

      1. un-div-inv 72.4%

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

      2. div-inv 72.4%

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

      3. metadata-eval 72.4%

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

      4. associate-*r/ 72.5%

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

      5. *-commutative 72.5%

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

      6. div-inv 72.5%

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

      7. associate-/l* 72.5%

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

   7. Applied egg-rr 72.5%

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

      1. expm1-log1p-u 68.8%

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

      2. expm1-udef 62.3%

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

      3. associate-/l/ 62.3%

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

      4. associate-/r/ 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. expm1-def 87.2%

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

       2. expm1-log1p 90.8%

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

       3. associate-/l* 90.7%

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

   11. Simplified 90.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{a \cdot b}{\pi}}\\ \end{array} \]

Alternative 11: 57.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. times-frac 76.7%

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

   2. *-commutative 76.7%

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

   3. times-frac 76.7%

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

   4. difference-of-squares 84.1%

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

   5. associate-/r* 85.4%

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

   6. metadata-eval 85.4%

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

   7. sub-neg 85.4%

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

   8. distribute-neg-frac 85.4%

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

   9. metadata-eval 85.4%

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

3. Simplified 85.4%

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

   1. div-inv 85.4%

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

5. Applied egg-rr 85.4%

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

   1. distribute-lft-in 82.2%

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

   2. associate-*l/ 82.2%

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

   3. un-div-inv 82.2%

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

   4. associate-*l/ 82.2%

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

   5. un-div-inv 82.2%

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

7. Applied egg-rr 82.2%

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

   1. metadata-eval 82.2%

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

   2. distribute-neg-frac 82.2%

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

   3. distribute-lft-out 85.4%

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

   4. associate-*l/ 85.4%

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

   5. associate-/r* 84.1%

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

   6. sub-neg 84.1%

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

   7. *-commutative 84.1%

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

   8. sub-neg 84.1%

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

   9. distribute-neg-frac 84.1%

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

   10. metadata-eval 84.1%

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

   11. *-commutative 84.1%

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

   12. times-frac 85.4%

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

   13. +-commutative 85.4%

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

9. Simplified 85.4%

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

    1. associate-*l/ 85.4%

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

11. Applied egg-rr 85.4%

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

12. Taylor expanded in a around inf 54.1%

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

    1. associate-/r* 54.1%

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

    2. unpow2 54.1%

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

14. Simplified 54.1%

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

15. Final simplification 54.1%

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

Reproduce

herbie shell --seed 2023238 
(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))))