NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 14.3s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. un-div-inv 80.2%

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

   2. difference-of-squares 92.0%

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

   3. associate-/r* 92.4%

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

   4. div-inv 92.4%

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

   5. metadata-eval 92.4%

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

4. Applied egg-rr 92.4%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-/l* 99.8%

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

   2. +-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{-1}{b} + \frac{2}{a}}{a}}{b - a}\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-135}:\\ \;\;\;\;\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{a + b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.4e+81)
   (/ (* (* PI 0.5) (/ (+ (/ -1.0 b) (/ 2.0 a)) a)) (- b a))
   (if (<= a -1.4e-135)
     (* (* PI 0.5) (/ (+ (/ 1.0 a) (/ -1.0 b)) (- (* b b) (* a a))))
     (/ (* (* PI 0.5) (/ (/ 1.0 a) (+ a b))) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.4e+81) {
		tmp = ((((double) M_PI) * 0.5) * (((-1.0 / b) + (2.0 / a)) / a)) / (b - a);
	} else if (a <= -1.4e-135) {
		tmp = (((double) M_PI) * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)));
	} else {
		tmp = ((((double) M_PI) * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.4e+81) {
		tmp = ((Math.PI * 0.5) * (((-1.0 / b) + (2.0 / a)) / a)) / (b - a);
	} else if (a <= -1.4e-135) {
		tmp = (Math.PI * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)));
	} else {
		tmp = ((Math.PI * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.4e+81:
		tmp = ((math.pi * 0.5) * (((-1.0 / b) + (2.0 / a)) / a)) / (b - a)
	elif a <= -1.4e-135:
		tmp = (math.pi * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)))
	else:
		tmp = ((math.pi * 0.5) * ((1.0 / a) / (a + b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.4e+81)
		tmp = Float64(Float64(Float64(pi * 0.5) * Float64(Float64(Float64(-1.0 / b) + Float64(2.0 / a)) / a)) / Float64(b - a));
	elseif (a <= -1.4e-135)
		tmp = Float64(Float64(pi * 0.5) * Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) * Float64(Float64(1.0 / a) / Float64(a + b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.4e+81)
		tmp = ((pi * 0.5) * (((-1.0 / b) + (2.0 / a)) / a)) / (b - a);
	elseif (a <= -1.4e-135)
		tmp = (pi * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)));
	else
		tmp = ((pi * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.4e+81], N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(N[(-1.0 / b), $MachinePrecision] + N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.4e-135], N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.40000000000000003e81

   1. Initial program 57.8%

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

      1. un-div-inv 57.8%

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

      2. difference-of-squares 89.1%

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

      3. associate-/r* 90.2%

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

      4. div-inv 90.2%

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

      5. metadata-eval 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l/ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*l/ 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in a around inf 100.0%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\frac{2 \cdot \frac{1}{a} - \frac{1}{b}}{a}}}{b - a} \]
   12. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\color{blue}{\frac{2 \cdot 1}{a}} - \frac{1}{b}}{a}}{b - a} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{\color{blue}{2}}{a} - \frac{1}{b}}{a}}{b - a} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\frac{\frac{2}{a} - \frac{1}{b}}{a}}}{b - a} \]

   if -3.40000000000000003e81 < a < -1.40000000000000012e-135

   1. Initial program 97.6%

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

      1. associate-*l* 97.7%

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

      2. *-rgt-identity 97.7%

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

      3. associate-/l* 97.7%

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

      4. metadata-eval 97.7%

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

      5. associate-*l/ 97.8%

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

      6. *-lft-identity 97.8%

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

      7. sub-neg 97.8%

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

      8. distribute-neg-frac 97.8%

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

      9. metadata-eval 97.8%

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

   3. Simplified 97.8%

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

   if -1.40000000000000012e-135 < a

   1. Initial program 82.0%

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

      1. un-div-inv 82.0%

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

      2. difference-of-squares 91.0%

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

      3. associate-/r* 91.4%

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

      4. div-inv 91.4%

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

      5. metadata-eval 91.4%

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

   4. Applied egg-rr 91.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around 0 81.9%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\color{blue}{\frac{1}{a}}}{a + b}}{b - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{-1}{b} + \frac{2}{a}}{a}}{b - a}\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-135}:\\ \;\;\;\;\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{a + b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)}{b - a}\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{\frac{\pi \cdot 0.5}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b}}{\left(b - a\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.5e-89)
   (/ (* (/ -1.0 b) (* 0.5 (/ PI a))) (- b a))
   (if (<= a -1.4e-281)
     (* (/ 1.0 a) (/ (/ (* PI 0.5) (+ a b)) (- b a)))
     (/ (/ (/ PI a) b) (* (- b a) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-89) {
		tmp = ((-1.0 / b) * (0.5 * (((double) M_PI) / a))) / (b - a);
	} else if (a <= -1.4e-281) {
		tmp = (1.0 / a) * (((((double) M_PI) * 0.5) / (a + b)) / (b - a));
	} else {
		tmp = ((((double) M_PI) / a) / b) / ((b - a) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-89) {
		tmp = ((-1.0 / b) * (0.5 * (Math.PI / a))) / (b - a);
	} else if (a <= -1.4e-281) {
		tmp = (1.0 / a) * (((Math.PI * 0.5) / (a + b)) / (b - a));
	} else {
		tmp = ((Math.PI / a) / b) / ((b - a) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.5e-89:
		tmp = ((-1.0 / b) * (0.5 * (math.pi / a))) / (b - a)
	elif a <= -1.4e-281:
		tmp = (1.0 / a) * (((math.pi * 0.5) / (a + b)) / (b - a))
	else:
		tmp = ((math.pi / a) / b) / ((b - a) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.5e-89)
		tmp = Float64(Float64(Float64(-1.0 / b) * Float64(0.5 * Float64(pi / a))) / Float64(b - a));
	elseif (a <= -1.4e-281)
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(Float64(pi * 0.5) / Float64(a + b)) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) / Float64(Float64(b - a) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.5e-89)
		tmp = ((-1.0 / b) * (0.5 * (pi / a))) / (b - a);
	elseif (a <= -1.4e-281)
		tmp = (1.0 / a) * (((pi * 0.5) / (a + b)) / (b - a));
	else
		tmp = ((pi / a) / b) / ((b - a) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.5e-89], N[(N[(N[(-1.0 / b), $MachinePrecision] * N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.4e-281], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.50000000000000012e-89

   1. Initial program 74.8%

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

      1. un-div-inv 74.8%

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

      2. difference-of-squares 92.6%

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

      3. associate-/r* 93.3%

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

      4. div-inv 93.3%

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

      5. metadata-eval 93.3%

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

   4. Applied egg-rr 93.3%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 80.6%

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

   8. Taylor expanded in a around inf 89.8%

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

   if -5.50000000000000012e-89 < a < -1.40000000000000003e-281

   1. Initial program 86.0%

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

      1. un-div-inv 85.9%

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

      2. difference-of-squares 93.8%

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

      3. associate-/r* 93.9%

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

      4. div-inv 93.9%

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

      5. metadata-eval 93.9%

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in a around 0 88.6%

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

   if -1.40000000000000003e-281 < a

   1. Initial program 82.3%

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

      1. associate-*l* 82.2%

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

   3. Simplified 82.2%

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

      1. associate-*l/ 82.3%

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

      2. difference-of-squares 90.9%

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

      3. sub-neg 90.9%

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

      4. *-un-lft-identity 90.9%

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

      5. neg-mul-1 90.9%

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

      6. div-inv 90.9%

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

      7. associate-/r* 99.7%

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

      8. add-sqr-sqrt 47.6%

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

      9. sqrt-unprod 79.9%

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

      10. frac-times 79.8%

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

      11. metadata-eval 79.8%

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

      12. metadata-eval 79.8%

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

      13. frac-times 79.9%

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

      14. sqrt-unprod 37.7%

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

      15. add-sqr-sqrt 75.4%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in a around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. associate-/r* 75.3%

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

   9. Simplified 75.3%

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

       1. frac-times 75.3%

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

       2. associate-/l/ 75.3%

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

   11. Applied egg-rr 75.3%

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

       1. associate-*r/ 75.3%

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

       2. *-rgt-identity 75.3%

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

       3. associate-/r* 75.3%

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

   13. Simplified 75.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)}{b - a}\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{\frac{\pi \cdot 0.5}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b}}{\left(b - a\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{a + b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.35e-89)
   (/ (* (/ -1.0 b) (* 0.5 (/ PI a))) (- b a))
   (/ (* (* PI 0.5) (/ (/ 1.0 a) (+ a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.35e-89) {
		tmp = ((-1.0 / b) * (0.5 * (((double) M_PI) / a))) / (b - a);
	} else {
		tmp = ((((double) M_PI) * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.35e-89) {
		tmp = ((-1.0 / b) * (0.5 * (Math.PI / a))) / (b - a);
	} else {
		tmp = ((Math.PI * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.35e-89:
		tmp = ((-1.0 / b) * (0.5 * (math.pi / a))) / (b - a)
	else:
		tmp = ((math.pi * 0.5) * ((1.0 / a) / (a + b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.35e-89)
		tmp = Float64(Float64(Float64(-1.0 / b) * Float64(0.5 * Float64(pi / a))) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) * Float64(Float64(1.0 / a) / Float64(a + b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.35e-89)
		tmp = ((-1.0 / b) * (0.5 * (pi / a))) / (b - a);
	else
		tmp = ((pi * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.35e-89], N[(N[(N[(-1.0 / b), $MachinePrecision] * N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.34999999999999998e-89

   1. Initial program 74.8%

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

      1. un-div-inv 74.8%

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

      2. difference-of-squares 92.6%

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

      3. associate-/r* 93.3%

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

      4. div-inv 93.3%

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

      5. metadata-eval 93.3%

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

   4. Applied egg-rr 93.3%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 80.6%

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

   8. Taylor expanded in a around inf 89.8%

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

   if -2.34999999999999998e-89 < a

   1. Initial program 83.2%

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

      1. un-div-inv 83.2%

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

      2. difference-of-squares 91.6%

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

      3. associate-/r* 91.9%

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

      4. div-inv 91.9%

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

      5. metadata-eval 91.9%

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

   4. Applied egg-rr 91.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around 0 81.8%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\color{blue}{\frac{1}{a}}}{a + b}}{b - a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{a + b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \frac{\pi \cdot 0.5}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{a + b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95e-78)
   (/ (* (/ -1.0 b) (/ (* PI 0.5) (+ a b))) (- b a))
   (/ (* (* PI 0.5) (/ (/ 1.0 a) (+ a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95e-78) {
		tmp = ((-1.0 / b) * ((((double) M_PI) * 0.5) / (a + b))) / (b - a);
	} else {
		tmp = ((((double) M_PI) * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.95e-78) {
		tmp = ((-1.0 / b) * ((Math.PI * 0.5) / (a + b))) / (b - a);
	} else {
		tmp = ((Math.PI * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.95e-78:
		tmp = ((-1.0 / b) * ((math.pi * 0.5) / (a + b))) / (b - a)
	else:
		tmp = ((math.pi * 0.5) * ((1.0 / a) / (a + b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95e-78)
		tmp = Float64(Float64(Float64(-1.0 / b) * Float64(Float64(pi * 0.5) / Float64(a + b))) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) * Float64(Float64(1.0 / a) / Float64(a + b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.95e-78)
		tmp = ((-1.0 / b) * ((pi * 0.5) / (a + b))) / (b - a);
	else
		tmp = ((pi * 0.5) * ((1.0 / a) / (a + b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95e-78], N[(N[(N[(-1.0 / b), $MachinePrecision] * N[(N[(Pi * 0.5), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9500000000000001e-78

   1. Initial program 78.3%

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

      1. un-div-inv 78.4%

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

      2. difference-of-squares 90.3%

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

      3. associate-/r* 90.9%

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

      4. div-inv 90.9%

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

      5. metadata-eval 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around inf 80.0%

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

   if 1.9500000000000001e-78 < b

   1. Initial program 83.4%

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

      1. un-div-inv 83.3%

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

      2. difference-of-squares 94.8%

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

      3. associate-/r* 94.9%

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

      4. div-inv 94.9%

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

      5. metadata-eval 94.9%

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

   4. Applied egg-rr 94.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around 0 91.7%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\color{blue}{\frac{1}{a}}}{a + b}}{b - a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \frac{\pi \cdot 0.5}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{a + b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. un-div-inv 80.2%

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

   2. difference-of-squares 92.0%

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

   3. associate-/r* 92.4%

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

   4. div-inv 92.4%

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

   5. metadata-eval 92.4%

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

4. Applied egg-rr 92.4%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* 99.7%

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

   2. *-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. sub-neg 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. metadata-eval 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 7: 75.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b}}{\left(b - a\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e-78)
   (/ (* (/ -1.0 b) (* 0.5 (/ PI a))) (- b a))
   (/ (/ (/ PI a) b) (* (- b a) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-78) {
		tmp = ((-1.0 / b) * (0.5 * (((double) M_PI) / a))) / (b - a);
	} else {
		tmp = ((((double) M_PI) / a) / b) / ((b - a) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-78) {
		tmp = ((-1.0 / b) * (0.5 * (Math.PI / a))) / (b - a);
	} else {
		tmp = ((Math.PI / a) / b) / ((b - a) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.8e-78:
		tmp = ((-1.0 / b) * (0.5 * (math.pi / a))) / (b - a)
	else:
		tmp = ((math.pi / a) / b) / ((b - a) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.8e-78)
		tmp = Float64(Float64(Float64(-1.0 / b) * Float64(0.5 * Float64(pi / a))) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) / Float64(Float64(b - a) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.8e-78)
		tmp = ((-1.0 / b) * (0.5 * (pi / a))) / (b - a);
	else
		tmp = ((pi / a) / b) / ((b - a) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.8e-78], N[(N[(N[(-1.0 / b), $MachinePrecision] * N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7999999999999999e-78

   1. Initial program 78.3%

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

      1. un-div-inv 78.4%

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

      2. difference-of-squares 90.3%

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

      3. associate-/r* 90.9%

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

      4. div-inv 90.9%

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

      5. metadata-eval 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 69.0%

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

   8. Taylor expanded in a around inf 74.0%

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

   if 3.7999999999999999e-78 < b

   1. Initial program 83.4%

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

      1. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

      1. associate-*l/ 83.3%

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

      2. difference-of-squares 94.8%

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

      3. sub-neg 94.8%

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

      4. *-un-lft-identity 94.8%

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

      5. neg-mul-1 94.8%

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

      6. div-inv 94.8%

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

      7. associate-/r* 99.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 91.6%

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

      10. frac-times 91.6%

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

      11. metadata-eval 91.6%

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

      12. metadata-eval 91.6%

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

      13. frac-times 91.6%

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

      14. sqrt-unprod 91.6%

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

      15. add-sqr-sqrt 91.6%

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

   6. Applied egg-rr 91.6%

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

   7. Taylor expanded in a around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-/r* 91.6%

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

   9. Simplified 91.6%

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

       1. frac-times 91.7%

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

       2. associate-/l/ 91.6%

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

   11. Applied egg-rr 91.6%

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

       1. associate-*r/ 91.6%

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

       2. *-rgt-identity 91.6%

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

       3. associate-/r* 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{-1}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b}}{\left(b - a\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot -0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.2e-78)
   (/ (* (/ (/ PI a) b) -0.5) (- b a))
   (/ (* (/ PI b) (/ 0.5 a)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.2e-78) {
		tmp = (((((double) M_PI) / a) / b) * -0.5) / (b - a);
	} else {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.2e-78) {
		tmp = (((Math.PI / a) / b) * -0.5) / (b - a);
	} else {
		tmp = ((Math.PI / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.2e-78:
		tmp = (((math.pi / a) / b) * -0.5) / (b - a)
	else:
		tmp = ((math.pi / b) * (0.5 / a)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.2e-78)
		tmp = Float64(Float64(Float64(Float64(pi / a) / b) * -0.5) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.2e-78)
		tmp = (((pi / a) / b) * -0.5) / (b - a);
	else
		tmp = ((pi / b) * (0.5 / a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.2e-78], N[(N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * -0.5), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2000000000000002e-78

   1. Initial program 78.3%

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

      1. un-div-inv 78.4%

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

      2. difference-of-squares 90.3%

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

      3. associate-/r* 90.9%

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

      4. div-inv 90.9%

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

      5. metadata-eval 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in a around inf 74.1%

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

       1. *-commutative 74.1%

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

       2. associate-/r* 74.0%

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

   13. Simplified 74.0%

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

   if 5.2000000000000002e-78 < b

   1. Initial program 83.4%

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

      1. associate-*l* 83.3%

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

      2. *-rgt-identity 83.3%

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

      3. associate-/l* 83.3%

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

      4. metadata-eval 83.3%

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

      5. associate-*l/ 83.3%

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

      6. *-lft-identity 83.3%

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

      7. sub-neg 83.3%

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

      8. distribute-neg-frac 83.3%

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

      9. metadata-eval 83.3%

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

   3. Simplified 83.3%

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

      1. metadata-eval 83.3%

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

      2. div-inv 83.3%

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

      3. associate-*r/ 83.3%

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

      4. *-commutative 83.3%

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

      5. difference-of-squares 94.8%

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

      6. associate-/r* 99.7%

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

   6. Applied egg-rr 91.7%

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

   7. Taylor expanded in a around 0 91.6%

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

      1. associate-*r/ 91.6%

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

   9. Simplified 91.6%

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

       1. times-frac 91.7%

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

   11. Applied egg-rr 91.7%

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

       1. *-commutative 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot -0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.5e-78)
   (/ (/ (* PI -0.5) (* a b)) (- b a))
   (/ (* (/ PI b) (/ 0.5 a)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-78) {
		tmp = ((((double) M_PI) * -0.5) / (a * b)) / (b - a);
	} else {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-78) {
		tmp = ((Math.PI * -0.5) / (a * b)) / (b - a);
	} else {
		tmp = ((Math.PI / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.5e-78:
		tmp = ((math.pi * -0.5) / (a * b)) / (b - a)
	else:
		tmp = ((math.pi / b) * (0.5 / a)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.5e-78)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(a * b)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.5e-78)
		tmp = ((pi * -0.5) / (a * b)) / (b - a);
	else
		tmp = ((pi / b) * (0.5 / a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.5e-78], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.4999999999999997e-78

   1. Initial program 78.3%

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

      1. un-div-inv 78.4%

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

      2. difference-of-squares 90.3%

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

      3. associate-/r* 90.9%

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

      4. div-inv 90.9%

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

      5. metadata-eval 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 74.1%

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

      1. associate-*r/ 74.1%

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

   9. Simplified 74.1%

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

   if 9.4999999999999997e-78 < b

   1. Initial program 83.4%

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

      1. associate-*l* 83.3%

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

      2. *-rgt-identity 83.3%

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

      3. associate-/l* 83.3%

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

      4. metadata-eval 83.3%

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

      5. associate-*l/ 83.3%

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

      6. *-lft-identity 83.3%

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

      7. sub-neg 83.3%

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

      8. distribute-neg-frac 83.3%

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

      9. metadata-eval 83.3%

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

   3. Simplified 83.3%

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

      1. metadata-eval 83.3%

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

      2. div-inv 83.3%

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

      3. associate-*r/ 83.3%

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

      4. *-commutative 83.3%

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

      5. difference-of-squares 94.8%

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

      6. associate-/r* 99.7%

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

   6. Applied egg-rr 91.7%

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

   7. Taylor expanded in a around 0 91.6%

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

      1. associate-*r/ 91.6%

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

   9. Simplified 91.6%

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

       1. times-frac 91.7%

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

   11. Applied egg-rr 91.7%

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

       1. *-commutative 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b}}{\left(b - a\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.8e-78)
   (/ (/ (* PI -0.5) (* a b)) (- b a))
   (/ (/ (/ PI a) b) (* (- b a) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.8e-78) {
		tmp = ((((double) M_PI) * -0.5) / (a * b)) / (b - a);
	} else {
		tmp = ((((double) M_PI) / a) / b) / ((b - a) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7.8e-78) {
		tmp = ((Math.PI * -0.5) / (a * b)) / (b - a);
	} else {
		tmp = ((Math.PI / a) / b) / ((b - a) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.8e-78:
		tmp = ((math.pi * -0.5) / (a * b)) / (b - a)
	else:
		tmp = ((math.pi / a) / b) / ((b - a) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.8e-78)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(a * b)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) / Float64(Float64(b - a) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7.8e-78)
		tmp = ((pi * -0.5) / (a * b)) / (b - a);
	else
		tmp = ((pi / a) / b) / ((b - a) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.8e-78], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.8000000000000004e-78

   1. Initial program 78.3%

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

      1. un-div-inv 78.4%

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

      2. difference-of-squares 90.3%

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

      3. associate-/r* 90.9%

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

      4. div-inv 90.9%

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

      5. metadata-eval 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 74.1%

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

      1. associate-*r/ 74.1%

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

   9. Simplified 74.1%

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

   if 7.8000000000000004e-78 < b

   1. Initial program 83.4%

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

      1. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

      1. associate-*l/ 83.3%

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

      2. difference-of-squares 94.8%

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

      3. sub-neg 94.8%

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

      4. *-un-lft-identity 94.8%

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

      5. neg-mul-1 94.8%

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

      6. div-inv 94.8%

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

      7. associate-/r* 99.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 91.6%

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

      10. frac-times 91.6%

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

      11. metadata-eval 91.6%

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

      12. metadata-eval 91.6%

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

      13. frac-times 91.6%

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

      14. sqrt-unprod 91.6%

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

      15. add-sqr-sqrt 91.6%

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

   6. Applied egg-rr 91.6%

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

   7. Taylor expanded in a around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-/r* 91.6%

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

   9. Simplified 91.6%

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

       1. frac-times 91.7%

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

       2. associate-/l/ 91.6%

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

   11. Applied egg-rr 91.6%

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

       1. associate-*r/ 91.6%

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

       2. *-rgt-identity 91.6%

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

       3. associate-/r* 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b}}{\left(b - a\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. *-rgt-identity 80.2%

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

   3. associate-/l* 80.2%

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

   4. metadata-eval 80.2%

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

   5. associate-*l/ 80.2%

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

   6. *-lft-identity 80.2%

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

   7. sub-neg 80.2%

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

   8. distribute-neg-frac 80.2%

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

   9. metadata-eval 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

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

   2. div-inv 80.2%

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

   3. associate-*r/ 80.2%

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

   4. *-commutative 80.2%

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

   5. difference-of-squares 91.9%

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

   6. associate-/r* 99.7%

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

6. Applied egg-rr 72.0%

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

7. Taylor expanded in a around 0 72.0%

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

   1. associate-*r/ 72.0%

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

9. Simplified 72.0%

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

    1. *-un-lft-identity 72.0%

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

    2. associate-/l/ 71.8%

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

    3. *-commutative 71.8%

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

    4. *-commutative 71.8%

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

11. Applied egg-rr 71.8%

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

    1. *-lft-identity 71.8%

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

    2. associate-/l* 71.8%

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

    3. *-commutative 71.8%

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

    4. *-commutative 71.8%

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

13. Simplified 71.8%

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

    1. pow1 71.8%

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

    2. associate-*l* 68.2%

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

15. Applied egg-rr 68.2%

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

    1. unpow1 68.2%

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

17. Simplified 68.2%

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

18. Final simplification 68.2%

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

Alternative 12: 65.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. *-rgt-identity 80.2%

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

   3. associate-/l* 80.2%

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

   4. metadata-eval 80.2%

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

   5. associate-*l/ 80.2%

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

   6. *-lft-identity 80.2%

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

   7. sub-neg 80.2%

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

   8. distribute-neg-frac 80.2%

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

   9. metadata-eval 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

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

   2. div-inv 80.2%

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

   3. associate-*r/ 80.2%

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

   4. *-commutative 80.2%

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

   5. difference-of-squares 91.9%

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

   6. associate-/r* 99.7%

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

6. Applied egg-rr 72.0%

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

7. Taylor expanded in a around 0 72.0%

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

   1. associate-*r/ 72.0%

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

9. Simplified 72.0%

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

    1. *-un-lft-identity 72.0%

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

    2. associate-/l/ 71.8%

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

    3. *-commutative 71.8%

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

    4. *-commutative 71.8%

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

11. Applied egg-rr 71.8%

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

    1. *-lft-identity 71.8%

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

    2. associate-/l* 71.8%

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

    3. *-commutative 71.8%

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

    4. *-commutative 71.8%

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

13. Simplified 71.8%

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

14. Final simplification 71.8%

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

Alternative 13: 65.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. *-rgt-identity 80.2%

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

   3. associate-/l* 80.2%

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

   4. metadata-eval 80.2%

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

   5. associate-*l/ 80.2%

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

   6. *-lft-identity 80.2%

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

   7. sub-neg 80.2%

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

   8. distribute-neg-frac 80.2%

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

   9. metadata-eval 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

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

   2. div-inv 80.2%

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

   3. associate-*r/ 80.2%

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

   4. *-commutative 80.2%

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

   5. difference-of-squares 91.9%

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

   6. associate-/r* 99.7%

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

6. Applied egg-rr 72.0%

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

7. Taylor expanded in a around 0 72.0%

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

   1. associate-*r/ 72.0%

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

9. Simplified 72.0%

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

    1. *-un-lft-identity 72.0%

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

    2. associate-/l/ 71.8%

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

    3. *-commutative 71.8%

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

    4. *-commutative 71.8%

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

11. Applied egg-rr 71.8%

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

    1. *-lft-identity 71.8%

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

    2. associate-/l* 71.8%

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

    3. *-commutative 71.8%

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

    4. *-commutative 71.8%

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

13. Simplified 71.8%

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

    1. pow1 71.8%

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

    2. associate-/r* 71.9%

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

15. Applied egg-rr 71.9%

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

    1. unpow1 71.9%

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

    2. associate-*r/ 72.0%

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

17. Simplified 72.0%

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

18. Final simplification 72.0%

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

Alternative 14: 65.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. *-rgt-identity 80.2%

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

   3. associate-/l* 80.2%

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

   4. metadata-eval 80.2%

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

   5. associate-*l/ 80.2%

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

   6. *-lft-identity 80.2%

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

   7. sub-neg 80.2%

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

   8. distribute-neg-frac 80.2%

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

   9. metadata-eval 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

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

   2. div-inv 80.2%

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

   3. associate-*r/ 80.2%

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

   4. *-commutative 80.2%

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

   5. difference-of-squares 91.9%

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

   6. associate-/r* 99.7%

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

6. Applied egg-rr 72.0%

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

7. Taylor expanded in a around 0 72.0%

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

   1. associate-*r/ 72.0%

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

9. Simplified 72.0%

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

    1. times-frac 72.1%

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

11. Applied egg-rr 72.1%

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

    1. *-commutative 72.1%

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

13. Simplified 72.1%

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

14. Final simplification 72.1%

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

Reproduce

herbie shell --seed 2024067 
(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))))