NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.6%

Time: 14.6s

Alternatives: 7

Speedup: 1.6×

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: 78.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. un-div-inv 76.7%

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

   2. difference-of-squares 88.0%

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

   3. associate-/r* 88.7%

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

   4. div-inv 88.7%

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

   5. metadata-eval 88.7%

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

4. Applied egg-rr 88.7%

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

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e-74)
   (* (/ PI a) (/ 0.5 (* b a)))
   (/ (* 0.5 (/ PI (* b a))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-74) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	} else {
		tmp = (0.5 * (((double) M_PI) / (b * a))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-74) {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	} else {
		tmp = (0.5 * (Math.PI / (b * a))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.3e-74:
		tmp = (math.pi / a) * (0.5 / (b * a))
	else:
		tmp = (0.5 * (math.pi / (b * a))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.3e-74)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(b * a))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.3e-74)
		tmp = (pi / a) * (0.5 / (b * a));
	else
		tmp = (0.5 * (pi / (b * a))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.3e-74], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999998e-74

   1. Initial program 77.0%

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

      1. associate-*l* 77.0%

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

      2. *-rgt-identity 77.0%

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

      3. associate-/l* 77.0%

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

      4. metadata-eval 77.0%

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

      5. associate-*l/ 77.1%

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

      6. *-lft-identity 77.1%

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

      7. sub-neg 77.1%

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

      8. distribute-neg-frac 77.1%

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

      9. metadata-eval 77.1%

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

   3. Simplified 77.1%

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

      1. metadata-eval 77.1%

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

      2. div-inv 77.1%

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

      3. associate-*r/ 77.0%

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

      4. associate-*l/ 77.1%

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

      5. un-div-inv 77.0%

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

      6. associate-*l/ 77.0%

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

      7. div-inv 77.0%

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

      8. neg-mul-1 77.0%

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

      9. sub-neg 77.0%

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

      10. frac-sub 77.0%

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

      11. frac-times 77.0%

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

   6. Applied egg-rr 77.0%

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

   7. Taylor expanded in b around 0 70.9%

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

      1. div-inv 70.9%

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

      2. *-commutative 70.9%

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

   9. Applied egg-rr 70.9%

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

       1. associate-/r* 70.9%

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

       2. metadata-eval 70.9%

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

       3. *-commutative 70.9%

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

   11. Simplified 70.9%

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

   if 2.2999999999999998e-74 < b

   1. Initial program 75.9%

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

      1. associate-*l* 75.9%

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

      2. *-rgt-identity 75.9%

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

      3. associate-/l* 75.9%

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

      4. metadata-eval 75.9%

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

      5. associate-*l/ 75.9%

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

      6. *-lft-identity 75.9%

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

      7. sub-neg 75.9%

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

      8. distribute-neg-frac 75.9%

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

      9. metadata-eval 75.9%

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

   3. Simplified 75.9%

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

      1. metadata-eval 75.9%

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

      2. div-inv 75.9%

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

      3. associate-*r/ 75.9%

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

      4. *-commutative 75.9%

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

      5. difference-of-squares 89.0%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 88.1%

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

   7. Taylor expanded in a around 0 88.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot 2}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.5e-74)
   (* (/ PI a) (/ 0.5 (* b a)))
   (/ (/ (/ PI a) (* b 2.0)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-74) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	} else {
		tmp = ((((double) M_PI) / a) / (b * 2.0)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-74) {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	} else {
		tmp = ((Math.PI / a) / (b * 2.0)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.5e-74:
		tmp = (math.pi / a) * (0.5 / (b * a))
	else:
		tmp = ((math.pi / a) / (b * 2.0)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.5e-74)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * 2.0)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7.5e-74)
		tmp = (pi / a) * (0.5 / (b * a));
	else
		tmp = ((pi / a) / (b * 2.0)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.5e-74], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5e-74

   1. Initial program 77.0%

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

      1. associate-*l* 77.0%

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

      2. *-rgt-identity 77.0%

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

      3. associate-/l* 77.0%

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

      4. metadata-eval 77.0%

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

      5. associate-*l/ 77.1%

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

      6. *-lft-identity 77.1%

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

      7. sub-neg 77.1%

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

      8. distribute-neg-frac 77.1%

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

      9. metadata-eval 77.1%

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

   3. Simplified 77.1%

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

      1. metadata-eval 77.1%

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

      2. div-inv 77.1%

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

      3. associate-*r/ 77.0%

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

      4. associate-*l/ 77.1%

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

      5. un-div-inv 77.0%

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

      6. associate-*l/ 77.0%

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

      7. div-inv 77.0%

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

      8. neg-mul-1 77.0%

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

      9. sub-neg 77.0%

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

      10. frac-sub 77.0%

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

      11. frac-times 77.0%

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

   6. Applied egg-rr 77.0%

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

   7. Taylor expanded in b around 0 70.9%

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

      1. div-inv 70.9%

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

      2. *-commutative 70.9%

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

   9. Applied egg-rr 70.9%

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

       1. associate-/r* 70.9%

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

       2. metadata-eval 70.9%

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

       3. *-commutative 70.9%

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

   11. Simplified 70.9%

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

   if 7.5e-74 < b

   1. Initial program 75.9%

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

      1. associate-*l* 75.9%

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

   3. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. div-inv 75.9%

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

      3. sub-neg 75.9%

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

      4. neg-mul-1 75.9%

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

      5. div-inv 75.9%

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

      6. difference-of-squares 89.0%

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

      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 88.0%

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

      10. frac-times 88.0%

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

      11. metadata-eval 88.0%

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

      12. metadata-eval 88.0%

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

      13. frac-times 88.0%

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

      14. sqrt-unprod 88.0%

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

      15. add-sqr-sqrt 88.0%

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

   6. Applied egg-rr 88.0%

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

   7. Taylor expanded in a around 0 88.0%

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

      1. div-inv 88.0%

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

      2. metadata-eval 88.0%

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

      3. associate-*r/ 88.0%

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

      4. metadata-eval 88.0%

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

      5. div-inv 88.0%

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

      6. associate-/r* 88.0%

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

      7. frac-times 88.1%

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

      8. div-inv 88.0%

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

   9. Applied egg-rr 88.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b \cdot 2}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(b \cdot a\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e-56) (* (/ PI a) (/ 0.5 (* b a))) (/ (/ PI b) (* (* b a) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5e-56) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	} else {
		tmp = (((double) M_PI) / b) / ((b * a) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.5e-56) {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	} else {
		tmp = (Math.PI / b) / ((b * a) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.5e-56:
		tmp = (math.pi / a) * (0.5 / (b * a))
	else:
		tmp = (math.pi / b) / ((b * a) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.5e-56)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(b * a) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.5e-56)
		tmp = (pi / a) * (0.5 / (b * a));
	else
		tmp = (pi / b) / ((b * a) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.5e-56], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4999999999999997e-56

   1. Initial program 77.9%

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

      1. associate-*l* 77.9%

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

      2. *-rgt-identity 77.9%

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

      3. associate-/l* 77.9%

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

      4. metadata-eval 77.9%

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

      5. associate-*l/ 78.0%

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

      6. *-lft-identity 78.0%

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

      7. sub-neg 78.0%

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

      8. distribute-neg-frac 78.0%

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

      9. metadata-eval 78.0%

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

   3. Simplified 78.0%

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

      1. metadata-eval 78.0%

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

      2. div-inv 78.0%

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

      3. associate-*r/ 77.9%

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

      4. associate-*l/ 78.0%

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

      5. un-div-inv 77.9%

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

      6. associate-*l/ 77.9%

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

      7. div-inv 77.9%

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

      8. neg-mul-1 77.9%

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

      9. sub-neg 77.9%

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

      10. frac-sub 77.8%

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

      11. frac-times 77.9%

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

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in b around 0 71.8%

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

      1. div-inv 71.8%

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

      2. *-commutative 71.8%

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

   9. Applied egg-rr 71.8%

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

       1. associate-/r* 71.8%

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

       2. metadata-eval 71.8%

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

       3. *-commutative 71.8%

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

   11. Simplified 71.8%

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

   if 6.4999999999999997e-56 < b

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. *-rgt-identity 73.7%

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

      3. associate-/l* 73.7%

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

      4. metadata-eval 73.7%

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

      5. associate-*l/ 73.7%

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

      6. *-lft-identity 73.7%

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

      7. sub-neg 73.7%

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

      8. distribute-neg-frac 73.7%

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

      9. metadata-eval 73.7%

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

   3. Simplified 73.7%

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

      1. metadata-eval 73.7%

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

      2. div-inv 73.7%

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

      3. associate-*r/ 73.7%

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

      4. associate-*l/ 73.8%

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

      5. un-div-inv 73.7%

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

      6. associate-*l/ 73.7%

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

      7. div-inv 73.7%

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

      8. neg-mul-1 73.7%

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

      9. sub-neg 73.7%

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

      10. frac-sub 73.7%

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

      11. frac-times 73.6%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in b around inf 88.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(b \cdot a\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. un-div-inv 76.7%

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

   2. difference-of-squares 88.0%

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

   3. associate-/r* 88.7%

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

   4. div-inv 88.7%

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

   5. metadata-eval 88.7%

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

4. Applied egg-rr 88.7%

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

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in a around 0 99.7%

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

10. Final simplification 99.7%

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

Alternative 6: 63.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. associate-*l* 76.7%

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

   2. *-rgt-identity 76.7%

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

   3. associate-/l* 76.7%

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

   4. metadata-eval 76.7%

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

   5. associate-*l/ 76.7%

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

   6. *-lft-identity 76.7%

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

   7. sub-neg 76.7%

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

   8. distribute-neg-frac 76.7%

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

   9. metadata-eval 76.7%

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

3. Simplified 76.7%

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

   1. metadata-eval 76.7%

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

   2. div-inv 76.7%

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

   3. associate-*r/ 76.7%

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

   4. associate-*l/ 76.7%

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

   5. un-div-inv 76.6%

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

   6. associate-*l/ 76.6%

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

   7. div-inv 76.6%

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

   8. neg-mul-1 76.6%

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

   9. sub-neg 76.6%

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

   10. frac-sub 76.6%

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

   11. frac-times 76.6%

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

6. Applied egg-rr 76.7%

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

7. Taylor expanded in b around 0 64.2%

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

   1. div-inv 64.2%

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

   2. *-commutative 64.2%

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

9. Applied egg-rr 64.2%

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

    1. associate-/r* 64.2%

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

    2. metadata-eval 64.2%

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

    3. *-commutative 64.2%

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

11. Simplified 64.2%

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

12. Final simplification 64.2%

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

Alternative 7: 63.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. associate-*l* 76.7%

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

   2. *-rgt-identity 76.7%

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

   3. associate-/l* 76.7%

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

   4. metadata-eval 76.7%

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

   5. associate-*l/ 76.7%

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

   6. *-lft-identity 76.7%

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

   7. sub-neg 76.7%

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

   8. distribute-neg-frac 76.7%

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

   9. metadata-eval 76.7%

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

3. Simplified 76.7%

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

   1. metadata-eval 76.7%

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

   2. div-inv 76.7%

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

   3. associate-*r/ 76.7%

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

   4. associate-*l/ 76.7%

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

   5. un-div-inv 76.6%

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

   6. associate-*l/ 76.6%

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

   7. div-inv 76.6%

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

   8. neg-mul-1 76.6%

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

   9. sub-neg 76.6%

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

   10. frac-sub 76.6%

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

   11. frac-times 76.6%

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

6. Applied egg-rr 76.7%

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

7. Taylor expanded in b around 0 64.2%

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

   1. *-un-lft-identity 64.2%

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

   2. associate-/l/ 64.5%

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

   3. *-commutative 64.5%

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

9. Applied egg-rr 64.5%

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

    1. *-lft-identity 64.5%

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

    2. *-commutative 64.5%

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

    3. *-commutative 64.5%

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

11. Simplified 64.5%

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

12. Final simplification 64.5%

    \[\leadsto \frac{\pi}{a \cdot \left(\left(b \cdot a\right) \cdot 2\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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))))