NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.3% → 99.6%

Time: 10.1s

Alternatives: 16

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: 79.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. inv-pow 81.6%

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

   2. difference-of-squares 91.8%

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

   3. unpow-prod-down 91.9%

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

   4. inv-pow 91.9%

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

   5. inv-pow 91.9%

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

3. Applied egg-rr 91.9%

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

   1. associate-*r/ 92.0%

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

   2. *-rgt-identity 92.0%

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

   3. +-commutative 92.0%

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

5. Simplified 92.0%

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

   1. associate-*r/ 92.0%

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

   2. div-inv 92.0%

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

   3. metadata-eval 92.0%

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

7. Applied egg-rr 92.0%

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

   1. associate-*r/ 92.0%

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

   2. *-rgt-identity 92.0%

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

   3. *-commutative 92.0%

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

9. Simplified 92.0%

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

    1. associate-*l/ 99.7%

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

11. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.7%

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

13. Applied egg-rr 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{-1}{b}}{b - a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e-145)
   (/ (* (/ (* 0.5 PI) (+ a b)) (/ -1.0 b)) (- b a))
   (if (<= b 1.4e+64)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (/ (/ PI 2.0) (- (* b b) (* a a))))
     (* (/ (/ 0.5 a) b) (/ PI b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e-145) {
		tmp = (((0.5 * ((double) M_PI)) / (a + b)) * (-1.0 / b)) / (b - a);
	} else if (b <= 1.4e+64) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((((double) M_PI) / 2.0) / ((b * b) - (a * a)));
	} else {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2e-145) {
		tmp = (((0.5 * Math.PI) / (a + b)) * (-1.0 / b)) / (b - a);
	} else if (b <= 1.4e+64) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((Math.PI / 2.0) / ((b * b) - (a * a)));
	} else {
		tmp = ((0.5 / a) / b) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2e-145:
		tmp = (((0.5 * math.pi) / (a + b)) * (-1.0 / b)) / (b - a)
	elif b <= 1.4e+64:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((math.pi / 2.0) / ((b * b) - (a * a)))
	else:
		tmp = ((0.5 / a) / b) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e-145)
		tmp = Float64(Float64(Float64(Float64(0.5 * pi) / Float64(a + b)) * Float64(-1.0 / b)) / Float64(b - a));
	elseif (b <= 1.4e+64)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(pi / 2.0) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2e-145)
		tmp = (((0.5 * pi) / (a + b)) * (-1.0 / b)) / (b - a);
	elseif (b <= 1.4e+64)
		tmp = ((1.0 / a) + (-1.0 / b)) * ((pi / 2.0) / ((b * b) - (a * a)));
	else
		tmp = ((0.5 / a) / b) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e-145], N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e+64], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / 2.0), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.99999999999999983e-145

   1. Initial program 80.4%

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

      1. inv-pow 80.4%

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

      2. difference-of-squares 92.8%

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

      3. unpow-prod-down 93.0%

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

      4. inv-pow 93.0%

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

      5. inv-pow 93.0%

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

   3. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

      3. +-commutative 93.1%

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

   5. Simplified 93.1%

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

      1. associate-*r/ 93.0%

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

      2. div-inv 93.0%

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

      3. metadata-eval 93.0%

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

   7. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.0%

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

      2. *-rgt-identity 93.0%

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

      3. *-commutative 93.0%

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

   9. Simplified 93.0%

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

       1. associate-*l/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in a around inf 74.7%

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

   if 1.99999999999999983e-145 < b < 1.40000000000000012e64

   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. associate-*r/ 96.8%

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

      2. *-rgt-identity 96.8%

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

      3. sub-neg 96.8%

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

      4. distribute-neg-frac 96.8%

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

      5. metadata-eval 96.8%

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

   3. Simplified 96.8%

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

   if 1.40000000000000012e64 < b

   1. Initial program 76.4%

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

      1. inv-pow 76.4%

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

      2. difference-of-squares 85.7%

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

      3. unpow-prod-down 85.7%

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

      4. inv-pow 85.7%

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

      5. inv-pow 85.7%

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

   3. Applied egg-rr 85.7%

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

      1. associate-*r/ 85.7%

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

      2. *-rgt-identity 85.7%

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

      3. +-commutative 85.7%

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

   5. Simplified 85.7%

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

      1. associate-*r/ 85.7%

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

      2. div-inv 85.7%

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

      3. metadata-eval 85.7%

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

   7. Applied egg-rr 85.7%

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

      1. associate-*r/ 85.8%

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

      2. *-rgt-identity 85.8%

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

      3. *-commutative 85.8%

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

   9. Simplified 85.8%

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

   10. Taylor expanded in a around 0 85.7%

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

       1. associate-*r/ 85.7%

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

       2. times-frac 85.7%

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

       3. associate-*r/ 85.7%

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

       4. unpow2 85.7%

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

       5. times-frac 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{-1}{b}}{b - a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 3: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot \pi}{a + b}\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{t_0 \cdot \frac{-1}{b}}{b - a}\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-234}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{t_0}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a + b}{\pi}}}{a \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 PI) (+ a b))))
   (if (<= a -1.6e+22)
     (/ (* t_0 (/ -1.0 b)) (- b a))
     (if (<= a -1.55e-234)
       (* (+ (/ 1.0 a) (/ -1.0 b)) (/ t_0 (- b a)))
       (/ (/ 0.5 (/ (+ a b) PI)) (* a (- b a)))))))

C

double code(double a, double b) {
	double t_0 = (0.5 * ((double) M_PI)) / (a + b);
	double tmp;
	if (a <= -1.6e+22) {
		tmp = (t_0 * (-1.0 / b)) / (b - a);
	} else if (a <= -1.55e-234) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / (b - a));
	} else {
		tmp = (0.5 / ((a + b) / ((double) M_PI))) / (a * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (0.5 * Math.PI) / (a + b);
	double tmp;
	if (a <= -1.6e+22) {
		tmp = (t_0 * (-1.0 / b)) / (b - a);
	} else if (a <= -1.55e-234) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / (b - a));
	} else {
		tmp = (0.5 / ((a + b) / Math.PI)) / (a * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (0.5 * math.pi) / (a + b)
	tmp = 0
	if a <= -1.6e+22:
		tmp = (t_0 * (-1.0 / b)) / (b - a)
	elif a <= -1.55e-234:
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / (b - a))
	else:
		tmp = (0.5 / ((a + b) / math.pi)) / (a * (b - a))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(0.5 * pi) / Float64(a + b))
	tmp = 0.0
	if (a <= -1.6e+22)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / b)) / Float64(b - a));
	elseif (a <= -1.55e-234)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(t_0 / Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a + b) / pi)) / Float64(a * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (0.5 * pi) / (a + b);
	tmp = 0.0;
	if (a <= -1.6e+22)
		tmp = (t_0 * (-1.0 / b)) / (b - a);
	elseif (a <= -1.55e-234)
		tmp = ((1.0 / a) + (-1.0 / b)) * (t_0 / (b - a));
	else
		tmp = (0.5 / ((a + b) / pi)) / (a * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.6e+22], N[(N[(t$95$0 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.55e-234], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a + b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.6e22

   1. Initial program 73.9%

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

      1. inv-pow 73.9%

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

      2. difference-of-squares 89.2%

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

      3. unpow-prod-down 89.2%

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

      4. inv-pow 89.2%

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

      5. inv-pow 89.2%

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

   3. Applied egg-rr 89.2%

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

      1. associate-*r/ 89.2%

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

      2. *-rgt-identity 89.2%

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

      3. +-commutative 89.2%

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

   5. Simplified 89.2%

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

      1. associate-*r/ 89.2%

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

      2. div-inv 89.2%

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

      3. metadata-eval 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. associate-*r/ 89.2%

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

      2. *-rgt-identity 89.2%

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

      3. *-commutative 89.2%

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

   9. Simplified 89.2%

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

       1. associate-*l/ 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in a around inf 98.3%

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

   if -1.6e22 < a < -1.5500000000000001e-234

   1. Initial program 90.9%

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

      1. inv-pow 90.9%

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

      2. difference-of-squares 97.1%

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

      3. unpow-prod-down 97.0%

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

      4. inv-pow 97.0%

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

      5. inv-pow 97.0%

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

   3. Applied egg-rr 97.0%

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

      1. associate-*r/ 97.1%

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

      2. *-rgt-identity 97.1%

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

      3. +-commutative 97.1%

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

   5. Simplified 97.1%

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

      1. associate-*r/ 97.1%

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

      2. div-inv 97.1%

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

      3. metadata-eval 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. associate-*r/ 97.1%

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

      2. *-rgt-identity 97.1%

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

      3. *-commutative 97.1%

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

   9. Simplified 97.1%

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

   if -1.5500000000000001e-234 < a

   1. Initial program 81.7%

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

      1. inv-pow 81.7%

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

      2. difference-of-squares 91.1%

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

      3. unpow-prod-down 91.4%

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

      4. inv-pow 91.4%

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

      5. inv-pow 91.4%

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

   3. Applied egg-rr 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-rgt-identity 91.4%

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

      3. +-commutative 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in a around 0 75.5%

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

      1. expm1-log1p-u 60.6%

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

      2. expm1-udef 52.4%

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

      3. un-div-inv 52.4%

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

      4. associate-*r/ 52.4%

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

      5. div-inv 52.4%

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

      6. metadata-eval 52.4%

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

      7. *-commutative 52.4%

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

      8. div-inv 52.4%

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

   8. Applied egg-rr 52.4%

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

      1. expm1-def 60.5%

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

      2. expm1-log1p 75.6%

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

      3. associate-/l/ 81.1%

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

      4. associate-/l* 81.1%

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

   10. Simplified 81.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{-1}{b}}{b - a}\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-234}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{0.5 \cdot \pi}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a + b}{\pi}}}{a \cdot \left(b - a\right)}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. inv-pow 81.6%

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

   2. difference-of-squares 91.8%

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

   3. unpow-prod-down 91.9%

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

   4. inv-pow 91.9%

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

   5. inv-pow 91.9%

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

3. Applied egg-rr 91.9%

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

   1. associate-*r/ 92.0%

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

   2. *-rgt-identity 92.0%

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

   3. +-commutative 92.0%

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

5. Simplified 92.0%

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

   1. associate-*r/ 92.0%

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

   2. div-inv 92.0%

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

   3. metadata-eval 92.0%

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

7. Applied egg-rr 92.0%

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

   1. associate-*r/ 92.0%

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

   2. *-rgt-identity 92.0%

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

   3. *-commutative 92.0%

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

9. Simplified 92.0%

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

    1. associate-*l/ 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 5: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{-1}{b \cdot 2}}{a + b} \cdot \frac{\pi}{b - a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a + b}{\pi}}}{a \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.9e-144)
   (* (/ (/ -1.0 (* b 2.0)) (+ a b)) (/ PI (- b a)))
   (if (<= b 1.65e-74)
     (* (/ PI (* a a)) (/ 0.5 b))
     (/ (/ 0.5 (/ (+ a b) PI)) (* a (- b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.9e-144) {
		tmp = ((-1.0 / (b * 2.0)) / (a + b)) * (((double) M_PI) / (b - a));
	} else if (b <= 1.65e-74) {
		tmp = (((double) M_PI) / (a * a)) * (0.5 / b);
	} else {
		tmp = (0.5 / ((a + b) / ((double) M_PI))) / (a * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.9e-144) {
		tmp = ((-1.0 / (b * 2.0)) / (a + b)) * (Math.PI / (b - a));
	} else if (b <= 1.65e-74) {
		tmp = (Math.PI / (a * a)) * (0.5 / b);
	} else {
		tmp = (0.5 / ((a + b) / Math.PI)) / (a * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.9e-144:
		tmp = ((-1.0 / (b * 2.0)) / (a + b)) * (math.pi / (b - a))
	elif b <= 1.65e-74:
		tmp = (math.pi / (a * a)) * (0.5 / b)
	else:
		tmp = (0.5 / ((a + b) / math.pi)) / (a * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.9e-144)
		tmp = Float64(Float64(Float64(-1.0 / Float64(b * 2.0)) / Float64(a + b)) * Float64(pi / Float64(b - a)));
	elseif (b <= 1.65e-74)
		tmp = Float64(Float64(pi / Float64(a * a)) * Float64(0.5 / b));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a + b) / pi)) / Float64(a * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.9e-144)
		tmp = ((-1.0 / (b * 2.0)) / (a + b)) * (pi / (b - a));
	elseif (b <= 1.65e-74)
		tmp = (pi / (a * a)) * (0.5 / b);
	else
		tmp = (0.5 / ((a + b) / pi)) / (a * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.9e-144], N[(N[(N[(-1.0 / N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e-74], N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a + b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.9000000000000001e-144

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*l/ 80.4%

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

      3. associate-*r/ 80.4%

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

      4. associate-/l* 80.3%

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

      5. sub-neg 80.3%

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

      6. distribute-neg-frac 80.3%

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

      7. metadata-eval 80.3%

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

      8. associate-*r/ 80.3%

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

      9. *-rgt-identity 80.3%

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

      10. difference-of-squares 92.8%

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

      11. associate-/r* 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in a around inf 68.9%

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

      1. expm1-log1p-u 52.2%

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

      2. expm1-udef 48.2%

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

      3. associate-/r/ 48.2%

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

      4. associate-/l/ 48.2%

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

      5. +-commutative 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. expm1-def 52.2%

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

      2. expm1-log1p 68.9%

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

      3. associate-*r/ 68.9%

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

      4. *-commutative 68.9%

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

      5. times-frac 74.7%

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

      6. associate-/l/ 74.7%

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

   8. Simplified 74.7%

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

   if 4.9000000000000001e-144 < b < 1.64999999999999998e-74

   1. Initial program 99.5%

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

      1. inv-pow 99.5%

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

      2. difference-of-squares 99.5%

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

      3. unpow-prod-down 99.7%

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

      4. inv-pow 99.7%

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

      5. inv-pow 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in a around inf 89.1%

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

      1. associate-*r/ 89.1%

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

      2. *-commutative 89.1%

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

      3. times-frac 89.2%

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

      4. unpow2 89.2%

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

   8. Simplified 89.2%

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

   if 1.64999999999999998e-74 < b

   1. Initial program 82.1%

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

      1. inv-pow 82.1%

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

      2. difference-of-squares 88.6%

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

      3. unpow-prod-down 88.6%

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

      4. inv-pow 88.6%

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

      5. inv-pow 88.6%

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

   3. Applied egg-rr 88.6%

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

      1. associate-*r/ 88.6%

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

      2. *-rgt-identity 88.6%

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

      3. +-commutative 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in a around 0 87.2%

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

      1. expm1-log1p-u 70.0%

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

      2. expm1-udef 56.1%

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

      3. un-div-inv 56.1%

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

      4. associate-*r/ 56.1%

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

      5. div-inv 56.1%

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

      6. metadata-eval 56.1%

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

      7. *-commutative 56.1%

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

      8. div-inv 56.1%

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

   8. Applied egg-rr 56.1%

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

      1. expm1-def 70.0%

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

      2. expm1-log1p 87.3%

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

      3. associate-/l/ 97.1%

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

      4. associate-/l* 97.0%

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

   10. Simplified 97.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{-1}{b \cdot 2}}{a + b} \cdot \frac{\pi}{b - a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a + b}{\pi}}}{a \cdot \left(b - a\right)}\\ \end{array} \]

Alternative 6: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{-1}{b}}{b - a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a + b}{\pi}}}{a \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-140)
   (/ (* (/ (* 0.5 PI) (+ a b)) (/ -1.0 b)) (- b a))
   (if (<= b 6e-74)
     (* (/ PI (* a a)) (/ 0.5 b))
     (/ (/ 0.5 (/ (+ a b) PI)) (* a (- b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-140) {
		tmp = (((0.5 * ((double) M_PI)) / (a + b)) * (-1.0 / b)) / (b - a);
	} else if (b <= 6e-74) {
		tmp = (((double) M_PI) / (a * a)) * (0.5 / b);
	} else {
		tmp = (0.5 / ((a + b) / ((double) M_PI))) / (a * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-140) {
		tmp = (((0.5 * Math.PI) / (a + b)) * (-1.0 / b)) / (b - a);
	} else if (b <= 6e-74) {
		tmp = (Math.PI / (a * a)) * (0.5 / b);
	} else {
		tmp = (0.5 / ((a + b) / Math.PI)) / (a * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.1e-140:
		tmp = (((0.5 * math.pi) / (a + b)) * (-1.0 / b)) / (b - a)
	elif b <= 6e-74:
		tmp = (math.pi / (a * a)) * (0.5 / b)
	else:
		tmp = (0.5 / ((a + b) / math.pi)) / (a * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e-140)
		tmp = Float64(Float64(Float64(Float64(0.5 * pi) / Float64(a + b)) * Float64(-1.0 / b)) / Float64(b - a));
	elseif (b <= 6e-74)
		tmp = Float64(Float64(pi / Float64(a * a)) * Float64(0.5 / b));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(a + b) / pi)) / Float64(a * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.1e-140)
		tmp = (((0.5 * pi) / (a + b)) * (-1.0 / b)) / (b - a);
	elseif (b <= 6e-74)
		tmp = (pi / (a * a)) * (0.5 / b);
	else
		tmp = (0.5 / ((a + b) / pi)) / (a * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e-140], N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-74], N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(N[(a + b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.10000000000000017e-140

   1. Initial program 80.5%

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

      1. inv-pow 80.5%

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

      2. difference-of-squares 92.8%

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

      3. unpow-prod-down 93.0%

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

      4. inv-pow 93.0%

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

      5. inv-pow 93.0%

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

   3. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

      3. +-commutative 93.1%

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

   5. Simplified 93.1%

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

      1. associate-*r/ 93.1%

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

      2. div-inv 93.1%

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

      3. metadata-eval 93.1%

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

   7. Applied egg-rr 93.1%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

      3. *-commutative 93.1%

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

   9. Simplified 93.1%

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

       1. associate-*l/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in a around inf 74.9%

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

   if 2.10000000000000017e-140 < b < 6.00000000000000014e-74

   1. Initial program 99.5%

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

      1. inv-pow 99.5%

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

      2. difference-of-squares 99.5%

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

      3. unpow-prod-down 99.7%

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

      4. inv-pow 99.7%

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

      5. inv-pow 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in a around inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. *-commutative 88.0%

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

      3. times-frac 88.1%

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

      4. unpow2 88.1%

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

   8. Simplified 88.1%

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

   if 6.00000000000000014e-74 < b

   1. Initial program 82.1%

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

      1. inv-pow 82.1%

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

      2. difference-of-squares 88.6%

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

      3. unpow-prod-down 88.6%

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

      4. inv-pow 88.6%

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

      5. inv-pow 88.6%

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

   3. Applied egg-rr 88.6%

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

      1. associate-*r/ 88.6%

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

      2. *-rgt-identity 88.6%

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

      3. +-commutative 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in a around 0 87.2%

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

      1. expm1-log1p-u 70.0%

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

      2. expm1-udef 56.1%

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

      3. un-div-inv 56.1%

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

      4. associate-*r/ 56.1%

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

      5. div-inv 56.1%

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

      6. metadata-eval 56.1%

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

      7. *-commutative 56.1%

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

      8. div-inv 56.1%

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

   8. Applied egg-rr 56.1%

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

      1. expm1-def 70.0%

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

      2. expm1-log1p 87.3%

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

      3. associate-/l/ 97.1%

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

      4. associate-/l* 97.0%

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

   10. Simplified 97.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a + b} \cdot \frac{-1}{b}}{b - a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a + b}{\pi}}}{a \cdot \left(b - a\right)}\\ \end{array} \]

Alternative 7: 70.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e-73)
   (/ (* 0.5 PI) (* b (* a a)))
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e-73) {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * a));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5e-73) {
		tmp = (0.5 * Math.PI) / (b * (a * a));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e-73:
		tmp = (0.5 * math.pi) / (b * (a * a))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e-73)
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e-73)
		tmp = (0.5 * pi) / (b * (a * a));
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e-73], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9999999999999998e-73

   1. Initial program 81.4%

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

   2. Taylor expanded in b around 0 67.8%

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

      1. associate-*r/ 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 4.9999999999999998e-73 < b

   1. Initial program 82.1%

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

      1. inv-pow 82.1%

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

      2. difference-of-squares 88.6%

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

      3. unpow-prod-down 88.6%

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

      4. inv-pow 88.6%

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

      5. inv-pow 88.6%

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

   3. Applied egg-rr 88.6%

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

      1. associate-*r/ 88.6%

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

      2. *-rgt-identity 88.6%

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

      3. +-commutative 88.6%

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

   5. Simplified 88.6%

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

      1. associate-*r/ 88.6%

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

      2. div-inv 88.6%

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

      3. metadata-eval 88.6%

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

   7. Applied egg-rr 88.6%

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

      1. associate-*r/ 88.7%

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

      2. *-rgt-identity 88.7%

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

      3. *-commutative 88.7%

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

   9. Simplified 88.7%

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

       1. associate-*l/ 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around 0 97.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]

Alternative 8: 70.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{-1}{b}}{-2 \cdot \frac{a \cdot a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6e-74)
   (/ (/ -1.0 b) (* -2.0 (/ (* a a) PI)))
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-74) {
		tmp = (-1.0 / b) / (-2.0 * ((a * a) / ((double) M_PI)));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-74) {
		tmp = (-1.0 / b) / (-2.0 * ((a * a) / Math.PI));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.6e-74:
		tmp = (-1.0 / b) / (-2.0 * ((a * a) / math.pi))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6e-74)
		tmp = Float64(Float64(-1.0 / b) / Float64(-2.0 * Float64(Float64(a * a) / pi)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.6e-74)
		tmp = (-1.0 / b) / (-2.0 * ((a * a) / pi));
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6e-74], N[(N[(-1.0 / b), $MachinePrecision] / N[(-2.0 * N[(N[(a * a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5999999999999999e-74

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*l/ 81.4%

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

      3. associate-*r/ 81.4%

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

      4. associate-/l* 81.4%

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

      5. sub-neg 81.4%

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

      6. distribute-neg-frac 81.4%

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

      7. metadata-eval 81.4%

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

      8. associate-*r/ 81.4%

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

      9. *-rgt-identity 81.4%

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

      10. difference-of-squares 93.2%

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

      11. associate-/r* 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in b around 0 59.0%

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

      1. unpow2 59.0%

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

   6. Simplified 59.0%

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

   7. Taylor expanded in a around inf 67.3%

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

   if 1.5999999999999999e-74 < b

   1. Initial program 82.1%

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

      1. inv-pow 82.1%

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

      2. difference-of-squares 88.6%

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

      3. unpow-prod-down 88.6%

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

      4. inv-pow 88.6%

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

      5. inv-pow 88.6%

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

   3. Applied egg-rr 88.6%

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

      1. associate-*r/ 88.6%

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

      2. *-rgt-identity 88.6%

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

      3. +-commutative 88.6%

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

   5. Simplified 88.6%

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

      1. associate-*r/ 88.6%

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

      2. div-inv 88.6%

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

      3. metadata-eval 88.6%

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

   7. Applied egg-rr 88.6%

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

      1. associate-*r/ 88.7%

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

      2. *-rgt-identity 88.7%

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

      3. *-commutative 88.7%

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

   9. Simplified 88.7%

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

       1. associate-*l/ 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around 0 97.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{-1}{b}}{-2 \cdot \frac{a \cdot a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]

Alternative 9: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.6e-77)
   (/ (/ (* PI -0.5) (* a b)) (- b a))
   (/ (* PI (/ (/ 0.5 a) b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.6e-77) {
		tmp = ((((double) M_PI) * -0.5) / (a * b)) / (b - a);
	} else {
		tmp = (((double) M_PI) * ((0.5 / a) / b)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.6e-77) {
		tmp = ((Math.PI * -0.5) / (a * b)) / (b - a);
	} else {
		tmp = (Math.PI * ((0.5 / a) / b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.6e-77:
		tmp = ((math.pi * -0.5) / (a * b)) / (b - a)
	else:
		tmp = (math.pi * ((0.5 / a) / b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.6e-77)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(a * b)) / Float64(b - a));
	else
		tmp = Float64(Float64(pi * Float64(Float64(0.5 / a) / b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.6e-77)
		tmp = ((pi * -0.5) / (a * b)) / (b - a);
	else
		tmp = (pi * ((0.5 / a) / b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.6e-77], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.59999999999999982e-77

   1. Initial program 81.1%

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

      1. inv-pow 81.1%

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

      2. difference-of-squares 92.0%

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

      3. unpow-prod-down 91.9%

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

      4. inv-pow 91.9%

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

      5. inv-pow 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. +-commutative 92.0%

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

   5. Simplified 92.0%

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

      1. associate-*r/ 92.0%

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

      2. div-inv 92.0%

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

      3. metadata-eval 92.0%

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

   7. Applied egg-rr 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. *-commutative 92.0%

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

   9. Simplified 92.0%

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

       1. associate-*l/ 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in a around inf 85.7%

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

       1. associate-*r/ 85.7%

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

       2. *-commutative 85.7%

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

   14. Simplified 85.7%

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

   if -6.59999999999999982e-77 < a

   1. Initial program 81.9%

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

      1. inv-pow 81.9%

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

      2. difference-of-squares 91.7%

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

      3. unpow-prod-down 91.9%

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

      4. inv-pow 91.9%

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

      5. inv-pow 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. +-commutative 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in a around 0 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. times-frac 70.1%

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

      3. unpow2 70.1%

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

   8. Simplified 70.1%

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

      1. associate-*r/ 70.2%

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

   10. Applied egg-rr 70.2%

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

       1. times-frac 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. associate-*r/ 75.6%

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

   14. Simplified 75.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \]

Alternative 10: 64.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-77)
   (* (/ PI b) (/ 0.5 (* a a)))
   (* (/ 0.5 a) (/ PI (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-77) {
		tmp = (((double) M_PI) / b) * (0.5 / (a * a));
	} else {
		tmp = (0.5 / a) * (((double) M_PI) / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-77) {
		tmp = (Math.PI / b) * (0.5 / (a * a));
	} else {
		tmp = (0.5 / a) * (Math.PI / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.8e-77:
		tmp = (math.pi / b) * (0.5 / (a * a))
	else:
		tmp = (0.5 / a) * (math.pi / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-77)
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * a)));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e-77)
		tmp = (pi / b) * (0.5 / (a * a));
	else
		tmp = (0.5 / a) * (pi / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e-77], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.79999999999999966e-77

   1. Initial program 81.1%

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

      1. inv-pow 81.1%

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

      2. difference-of-squares 92.0%

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

      3. unpow-prod-down 91.9%

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

      4. inv-pow 91.9%

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

      5. inv-pow 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. +-commutative 92.0%

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

   5. Simplified 92.0%

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

      1. associate-*r/ 92.0%

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

      2. div-inv 92.0%

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

      3. metadata-eval 92.0%

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

   7. Applied egg-rr 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. *-commutative 92.0%

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

   9. Simplified 92.0%

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

       1. associate-*l/ 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in a around inf 72.9%

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

       1. associate-*r/ 72.9%

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

       2. times-frac 72.0%

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

       3. unpow2 72.0%

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

   14. Simplified 72.0%

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

   if -6.79999999999999966e-77 < a

   1. Initial program 81.9%

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

      1. inv-pow 81.9%

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

      2. difference-of-squares 91.7%

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

      3. unpow-prod-down 91.9%

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

      4. inv-pow 91.9%

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

      5. inv-pow 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. +-commutative 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in a around 0 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. times-frac 70.1%

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

      3. unpow2 70.1%

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

   8. Simplified 70.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 11: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e-65) (* (/ PI b) (/ 0.5 (* a a))) (* (/ PI b) (/ 0.5 (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e-65) {
		tmp = (((double) M_PI) / b) * (0.5 / (a * a));
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.7e-65) {
		tmp = (Math.PI / b) * (0.5 / (a * a));
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.7e-65:
		tmp = (math.pi / b) * (0.5 / (a * a))
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.7e-65)
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * a)));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.7e-65)
		tmp = (pi / b) * (0.5 / (a * a));
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.7e-65], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7e-65

   1. Initial program 81.2%

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

      1. inv-pow 81.2%

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

      2. difference-of-squares 92.8%

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

      3. unpow-prod-down 93.0%

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

      4. inv-pow 93.0%

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

      5. inv-pow 93.0%

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

   3. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.0%

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

      2. *-rgt-identity 93.0%

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

      3. +-commutative 93.0%

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

   5. Simplified 93.0%

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

      1. associate-*r/ 93.0%

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

      2. div-inv 93.0%

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

      3. metadata-eval 93.0%

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

   7. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.0%

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

      2. *-rgt-identity 93.0%

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

      3. *-commutative 93.0%

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

   9. Simplified 93.0%

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

       1. associate-*l/ 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in a around inf 67.3%

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

       1. associate-*r/ 67.3%

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

       2. times-frac 66.8%

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

       3. unpow2 66.8%

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

   14. Simplified 66.8%

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

   if 3.7e-65 < b

   1. Initial program 82.6%

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

      1. inv-pow 82.6%

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

      2. difference-of-squares 89.3%

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

      3. unpow-prod-down 89.3%

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

      4. inv-pow 89.3%

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

      5. inv-pow 89.3%

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

   3. Applied egg-rr 89.3%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. +-commutative 89.4%

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

   5. Simplified 89.4%

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

   6. Taylor expanded in a around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. times-frac 83.0%

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

      3. unpow2 83.0%

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

   8. Simplified 83.0%

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

      1. associate-*r/ 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. times-frac 93.3%

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

   12. Applied egg-rr 93.3%

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

       1. *-commutative 93.3%

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

       2. associate-/r* 93.2%

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

   14. Simplified 93.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 12: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.55e-65)
   (* (/ PI (* a a)) (/ 0.5 b))
   (* (/ PI b) (/ 0.5 (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.55e-65) {
		tmp = (((double) M_PI) / (a * a)) * (0.5 / b);
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.55e-65) {
		tmp = (Math.PI / (a * a)) * (0.5 / b);
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.55e-65:
		tmp = (math.pi / (a * a)) * (0.5 / b)
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.55e-65)
		tmp = Float64(Float64(pi / Float64(a * a)) * Float64(0.5 / b));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.55e-65)
		tmp = (pi / (a * a)) * (0.5 / b);
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.55e-65], N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.55000000000000014e-65

   1. Initial program 81.2%

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

      1. inv-pow 81.2%

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

      2. difference-of-squares 92.8%

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

      3. unpow-prod-down 93.0%

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

      4. inv-pow 93.0%

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

      5. inv-pow 93.0%

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

   3. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.0%

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

      2. *-rgt-identity 93.0%

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

      3. +-commutative 93.0%

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

   5. Simplified 93.0%

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

   6. Taylor expanded in a around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. *-commutative 67.3%

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

      3. times-frac 66.8%

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

      4. unpow2 66.8%

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

   8. Simplified 66.8%

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

   if 3.55000000000000014e-65 < b

   1. Initial program 82.6%

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

      1. inv-pow 82.6%

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

      2. difference-of-squares 89.3%

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

      3. unpow-prod-down 89.3%

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

      4. inv-pow 89.3%

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

      5. inv-pow 89.3%

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

   3. Applied egg-rr 89.3%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. +-commutative 89.4%

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

   5. Simplified 89.4%

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

   6. Taylor expanded in a around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. times-frac 83.0%

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

      3. unpow2 83.0%

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

   8. Simplified 83.0%

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

      1. associate-*r/ 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. times-frac 93.3%

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

   12. Applied egg-rr 93.3%

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

       1. *-commutative 93.3%

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

       2. associate-/r* 93.2%

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

   14. Simplified 93.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]

Alternative 13: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e-65) (* (/ PI (* a a)) (/ 0.5 b)) (* (/ (/ 0.5 a) b) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e-65) {
		tmp = (((double) M_PI) / (a * a)) * (0.5 / b);
	} else {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.7e-65) {
		tmp = (Math.PI / (a * a)) * (0.5 / b);
	} else {
		tmp = ((0.5 / a) / b) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.7e-65:
		tmp = (math.pi / (a * a)) * (0.5 / b)
	else:
		tmp = ((0.5 / a) / b) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.7e-65)
		tmp = Float64(Float64(pi / Float64(a * a)) * Float64(0.5 / b));
	else
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.7e-65)
		tmp = (pi / (a * a)) * (0.5 / b);
	else
		tmp = ((0.5 / a) / b) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.7e-65], N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7e-65

   1. Initial program 81.2%

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

      1. inv-pow 81.2%

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

      2. difference-of-squares 92.8%

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

      3. unpow-prod-down 93.0%

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

      4. inv-pow 93.0%

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

      5. inv-pow 93.0%

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

   3. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.0%

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

      2. *-rgt-identity 93.0%

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

      3. +-commutative 93.0%

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

   5. Simplified 93.0%

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

   6. Taylor expanded in a around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. *-commutative 67.3%

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

      3. times-frac 66.8%

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

      4. unpow2 66.8%

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

   8. Simplified 66.8%

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

   if 3.7e-65 < b

   1. Initial program 82.6%

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

      1. inv-pow 82.6%

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

      2. difference-of-squares 89.3%

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

      3. unpow-prod-down 89.3%

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

      4. inv-pow 89.3%

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

      5. inv-pow 89.3%

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

   3. Applied egg-rr 89.3%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. +-commutative 89.4%

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

   5. Simplified 89.4%

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

      1. associate-*r/ 89.4%

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

      2. div-inv 89.4%

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

      3. metadata-eval 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. *-commutative 89.4%

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

   9. Simplified 89.4%

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

   10. Taylor expanded in a around 0 83.1%

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

       1. associate-*r/ 83.1%

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

       2. times-frac 83.0%

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

       3. associate-*r/ 83.0%

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

       4. unpow2 83.0%

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

       5. times-frac 93.3%

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

   12. Simplified 93.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 14: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e-65) (/ (* 0.5 PI) (* b (* a a))) (* (/ (/ 0.5 a) b) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e-65) {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * a));
	} else {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.7e-65) {
		tmp = (0.5 * Math.PI) / (b * (a * a));
	} else {
		tmp = ((0.5 / a) / b) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.7e-65:
		tmp = (0.5 * math.pi) / (b * (a * a))
	else:
		tmp = ((0.5 / a) / b) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.7e-65)
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * a)));
	else
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.7e-65)
		tmp = (0.5 * pi) / (b * (a * a));
	else
		tmp = ((0.5 / a) / b) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.7e-65], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7e-65

   1. Initial program 81.2%

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

   2. Taylor expanded in b around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. unpow2 67.3%

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

   4. Simplified 67.3%

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

   if 3.7e-65 < b

   1. Initial program 82.6%

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

      1. inv-pow 82.6%

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

      2. difference-of-squares 89.3%

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

      3. unpow-prod-down 89.3%

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

      4. inv-pow 89.3%

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

      5. inv-pow 89.3%

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

   3. Applied egg-rr 89.3%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. +-commutative 89.4%

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

   5. Simplified 89.4%

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

      1. associate-*r/ 89.4%

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

      2. div-inv 89.4%

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

      3. metadata-eval 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. *-commutative 89.4%

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

   9. Simplified 89.4%

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

   10. Taylor expanded in a around 0 83.1%

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

       1. associate-*r/ 83.1%

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

       2. times-frac 83.0%

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

       3. associate-*r/ 83.0%

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

       4. unpow2 83.0%

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

       5. times-frac 93.3%

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

   12. Simplified 93.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 15: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.55e-65)
   (/ (* 0.5 PI) (* b (* a a)))
   (/ (* PI (/ (/ 0.5 a) b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.55e-65) {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * a));
	} else {
		tmp = (((double) M_PI) * ((0.5 / a) / b)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.55e-65) {
		tmp = (0.5 * Math.PI) / (b * (a * a));
	} else {
		tmp = (Math.PI * ((0.5 / a) / b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.55e-65:
		tmp = (0.5 * math.pi) / (b * (a * a))
	else:
		tmp = (math.pi * ((0.5 / a) / b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.55e-65)
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * a)));
	else
		tmp = Float64(Float64(pi * Float64(Float64(0.5 / a) / b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.55e-65)
		tmp = (0.5 * pi) / (b * (a * a));
	else
		tmp = (pi * ((0.5 / a) / b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.55e-65], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.55000000000000014e-65

   1. Initial program 81.2%

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

   2. Taylor expanded in b around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. unpow2 67.3%

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

   4. Simplified 67.3%

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

   if 3.55000000000000014e-65 < b

   1. Initial program 82.6%

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

      1. inv-pow 82.6%

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

      2. difference-of-squares 89.3%

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

      3. unpow-prod-down 89.3%

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

      4. inv-pow 89.3%

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

      5. inv-pow 89.3%

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

   3. Applied egg-rr 89.3%

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

      1. associate-*r/ 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. +-commutative 89.4%

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

   5. Simplified 89.4%

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

   6. Taylor expanded in a around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. times-frac 83.0%

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

      3. unpow2 83.0%

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

   8. Simplified 83.0%

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

      1. associate-*r/ 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. times-frac 93.3%

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

   12. Applied egg-rr 93.3%

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

       1. associate-*r/ 93.4%

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

   14. Simplified 93.4%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \]

Alternative 16: 57.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. inv-pow 81.6%

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

   2. difference-of-squares 91.8%

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

   3. unpow-prod-down 91.9%

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

   4. inv-pow 91.9%

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

   5. inv-pow 91.9%

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

3. Applied egg-rr 91.9%

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

   1. associate-*r/ 92.0%

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

   2. *-rgt-identity 92.0%

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

   3. +-commutative 92.0%

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

5. Simplified 92.0%

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

6. Taylor expanded in a around 0 64.4%

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

   1. associate-*r/ 64.4%

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

   2. times-frac 63.9%

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

   3. unpow2 63.9%

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

8. Simplified 63.9%

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

9. Final simplification 63.9%

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

Reproduce

herbie shell --seed 2023271 
(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))))