NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 10.6s

Alternatives: 9

Speedup: 1.9×

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: 78.2% 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, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. associate-*l/ N/A

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

   2. frac-sub N/A

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

   3. frac-times N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

   17. *-lowering-*.f64 80.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

4. Applied egg-rr 80.1%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot \left(b \cdot a\right)}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a} \cdot \left(b - a\right)\right)}\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot a}\right), \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot a}\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{b \cdot b - a \cdot a} \cdot \left(b - a\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot a\right)\right), \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   9. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(b - a\right)\right)}\right)\right) \]

   11. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{b - a}}}\right)\right) \]

   12. flip-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{b + \color{blue}{a}}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\frac{1}{b + a}\right)}\right)\right) \]

   14. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\color{blue}{1}}{b + a}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(1, \color{blue}{\left(b + a\right)}\right)\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(1, \left(a + \color{blue}{b}\right)\right)\right)\right) \]

   17. +-lowering-+.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a, \color{blue}{b}\right)\right)\right)\right) \]

6. Applied egg-rr 99.6%

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

Alternative 2: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.35e-65)
   (/ (* 0.5 PI) (* a (* a b)))
   (/ (/ PI b) (* (* a b) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.35e-65) {
		tmp = (0.5 * ((double) M_PI)) / (a * (a * b));
	} else {
		tmp = (((double) M_PI) / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.35e-65) {
		tmp = (0.5 * Math.PI) / (a * (a * b));
	} else {
		tmp = (Math.PI / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.35e-65:
		tmp = (0.5 * math.pi) / (a * (a * b))
	else:
		tmp = (math.pi / b) / ((a * b) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.35e-65)
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.35e-65)
		tmp = (0.5 * pi) / (a * (a * b));
	else
		tmp = (pi / b) / ((a * b) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.35e-65], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3499999999999999e-65

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left({a}^{2} \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right), \left(\color{blue}{{a}^{2}} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \left({a}^{\color{blue}{2}} \cdot b\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{b}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(a \cdot a\right), b\right)\right) \]

      7. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right) \]

   5. Simplified 78.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\left(a \cdot a\right) \cdot b\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(\left(\color{blue}{a} \cdot a\right) \cdot b\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \color{blue}{\left(b \cdot a\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 91.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   7. Applied egg-rr 91.1%

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

   if -1.3499999999999999e-65 < a

   1. Initial program 79.2%

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

      1. associate-*l/ N/A

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

      2. frac-sub N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      17. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

      2. PI-lowering-PI.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 72.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\pi}}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-67)
   (/ (* 0.5 PI) (* a (* a b)))
   (/ (/ 0.5 (/ b PI)) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-67) {
		tmp = (0.5 * ((double) M_PI)) / (a * (a * b));
	} else {
		tmp = (0.5 / (b / ((double) M_PI))) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-67) {
		tmp = (0.5 * Math.PI) / (a * (a * b));
	} else {
		tmp = (0.5 / (b / Math.PI)) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e-67:
		tmp = (0.5 * math.pi) / (a * (a * b))
	else:
		tmp = (0.5 / (b / math.pi)) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e-67)
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 / Float64(b / pi)) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e-67)
		tmp = (0.5 * pi) / (a * (a * b));
	else
		tmp = (0.5 / (b / pi)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e-67], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b / Pi), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.40000000000000032e-67

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left({a}^{2} \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right), \left(\color{blue}{{a}^{2}} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \left({a}^{\color{blue}{2}} \cdot b\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{b}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(a \cdot a\right), b\right)\right) \]

      7. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right) \]

   5. Simplified 78.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\left(a \cdot a\right) \cdot b\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(\left(\color{blue}{a} \cdot a\right) \cdot b\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \color{blue}{\left(b \cdot a\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 91.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   7. Applied egg-rr 91.1%

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

   if -5.40000000000000032e-67 < a

   1. Initial program 79.2%

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

      1. associate-*l/ N/A

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

      2. frac-sub N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      17. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

      2. PI-lowering-PI.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 72.4%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2}\right), \color{blue}{\left(a \cdot b\right)}\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b} \cdot \frac{1}{2}\right), \left(\color{blue}{a} \cdot b\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{b}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2}\right), \left(a \cdot b\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{b}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2}\right), \left(a \cdot b\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{\frac{b}{\mathsf{PI}\left(\right)}}\right), \left(\color{blue}{a} \cdot b\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{\frac{b}{\mathsf{PI}\left(\right)}}\right), \left(a \cdot b\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{b}{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{a} \cdot b\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(b, \mathsf{PI}\left(\right)\right)\right), \left(a \cdot b\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(b, \mathsf{PI.f64}\left(\right)\right)\right), \left(a \cdot b\right)\right) \]

      12. *-lowering-*.f64 72.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(b, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 72.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\pi}}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{\frac{\pi}{b}}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-65)
   (/ (* 0.5 PI) (* a (* a b)))
   (/ (/ 0.5 (/ a (/ PI b))) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-65) {
		tmp = (0.5 * ((double) M_PI)) / (a * (a * b));
	} else {
		tmp = (0.5 / (a / (((double) M_PI) / b))) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-65) {
		tmp = (0.5 * Math.PI) / (a * (a * b));
	} else {
		tmp = (0.5 / (a / (Math.PI / b))) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.3e-65:
		tmp = (0.5 * math.pi) / (a * (a * b))
	else:
		tmp = (0.5 / (a / (math.pi / b))) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.3e-65)
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 / Float64(a / Float64(pi / b))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e-65)
		tmp = (0.5 * pi) / (a * (a * b));
	else
		tmp = (0.5 / (a / (pi / b))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.3e-65], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a / N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3e-65

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left({a}^{2} \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right), \left(\color{blue}{{a}^{2}} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \left({a}^{\color{blue}{2}} \cdot b\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{b}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(a \cdot a\right), b\right)\right) \]

      7. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right) \]

   5. Simplified 78.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\left(a \cdot a\right) \cdot b\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(\left(\color{blue}{a} \cdot a\right) \cdot b\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \color{blue}{\left(b \cdot a\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 91.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   7. Applied egg-rr 91.1%

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

   if -2.3e-65 < a

   1. Initial program 79.2%

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

      1. associate-*l/ N/A

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

      2. frac-sub N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      17. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

      2. PI-lowering-PI.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 72.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2 \cdot \left(a \cdot b\right)}}{b} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot b}}{2}}{b} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot b}}{2}\right), \color{blue}{b}\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{2}{\frac{\mathsf{PI}\left(\right)}{a \cdot b}}}\right), b\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right), b\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right), b\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\frac{a \cdot b}{\mathsf{PI}\left(\right)}}\right), b\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{\frac{a \cdot b}{\mathsf{PI}\left(\right)}}\right), b\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a \cdot b}{\mathsf{PI}\left(\right)}\right)\right), b\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot \frac{b}{\mathsf{PI}\left(\right)}\right)\right), b\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{b}}\right)\right), b\right) \]

      14. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\frac{a}{\frac{\mathsf{PI}\left(\right)}{b}}\right)\right), b\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \left(\frac{\mathsf{PI}\left(\right)}{b}\right)\right)\right), b\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right)\right)\right), b\right) \]

      17. PI-lowering-PI.f64 72.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right)\right)\right), b\right) \]

   9. Applied egg-rr 72.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{\frac{\pi}{b}}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e-65)
   (/ (* 0.5 PI) (* a (* a b)))
   (* PI (/ (/ 0.5 (* a b)) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e-65) {
		tmp = (0.5 * ((double) M_PI)) / (a * (a * b));
	} else {
		tmp = ((double) M_PI) * ((0.5 / (a * b)) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.8e-65) {
		tmp = (0.5 * Math.PI) / (a * (a * b));
	} else {
		tmp = Math.PI * ((0.5 / (a * b)) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.8e-65:
		tmp = (0.5 * math.pi) / (a * (a * b))
	else:
		tmp = math.pi * ((0.5 / (a * b)) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e-65)
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(pi * Float64(Float64(0.5 / Float64(a * b)) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.8e-65)
		tmp = (0.5 * pi) / (a * (a * b));
	else
		tmp = pi * ((0.5 / (a * b)) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e-65], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8000000000000002e-65

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left({a}^{2} \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right), \left(\color{blue}{{a}^{2}} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \left({a}^{\color{blue}{2}} \cdot b\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{b}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(a \cdot a\right), b\right)\right) \]

      7. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right) \]

   5. Simplified 78.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\left(a \cdot a\right) \cdot b\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(\left(\color{blue}{a} \cdot a\right) \cdot b\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \color{blue}{\left(b \cdot a\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 91.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   7. Applied egg-rr 91.1%

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

   if -3.8000000000000002e-65 < a

   1. Initial program 79.2%

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

      1. associate-*l/ N/A

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

      2. frac-sub N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      17. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

      2. PI-lowering-PI.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 72.4%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

         \[\leadsto {\left(\left(2 \cdot \left(b \cdot a\right)\right) \cdot b\right)}^{-1} \cdot \mathsf{PI}\left(\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(b \cdot \left(2 \cdot \left(b \cdot a\right)\right)\right)}^{-1} \cdot \mathsf{PI}\left(\right) \]

      6. pow-prod-down N/A

         \[\leadsto \left({b}^{-1} \cdot {\left(2 \cdot \left(b \cdot a\right)\right)}^{-1}\right) \cdot \mathsf{PI}\left(\right) \]

      7. inv-pow N/A

         \[\leadsto \left(\frac{1}{b} \cdot {\left(2 \cdot \left(b \cdot a\right)\right)}^{-1}\right) \cdot \mathsf{PI}\left(\right) \]

      8. inv-pow N/A

         \[\leadsto \left(\frac{1}{b} \cdot \frac{1}{2 \cdot \left(b \cdot a\right)}\right) \cdot \mathsf{PI}\left(\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{1}{2 \cdot \left(b \cdot a\right)}\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{1}{2 \cdot \left(a \cdot b\right)}\right), \mathsf{PI}\left(\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{\frac{1}{2}}{a \cdot b}\right), \mathsf{PI}\left(\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{\frac{1}{2}}{a \cdot b}\right), \mathsf{PI}\left(\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot \frac{\frac{1}{2}}{a \cdot b}}{b}\right), \mathsf{PI}\left(\right)\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{1}{2}}{a \cdot b}}{b}\right), \mathsf{PI}\left(\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{a \cdot b}\right), b\right), \mathsf{PI}\left(\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot b\right)\right), b\right), \mathsf{PI}\left(\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), b\right), \mathsf{PI}\left(\right)\right) \]

      18. PI-lowering-PI.f64 72.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), b\right), \mathsf{PI.f64}\left(\right)\right) \]

   9. Applied egg-rr 72.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.2e-65)
   (* (/ 0.5 a) (/ PI (* a b)))
   (* PI (/ (/ 0.5 (* a b)) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.2e-65) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = ((double) M_PI) * ((0.5 / (a * b)) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7.2e-65) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = Math.PI * ((0.5 / (a * b)) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7.2e-65:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = math.pi * ((0.5 / (a * b)) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.2e-65)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(pi * Float64(Float64(0.5 / Float64(a * b)) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.2e-65)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = pi * ((0.5 / (a * b)) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.2e-65], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.1999999999999996e-65

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left({a}^{2} \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right), \left(\color{blue}{{a}^{2}} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \left({a}^{\color{blue}{2}} \cdot b\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{b}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(a \cdot a\right), b\right)\right) \]

      7. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right) \]

   5. Simplified 78.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b \cdot a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{b \cdot a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot a\right)}\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} \cdot a\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 90.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   7. Applied egg-rr 90.2%

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

   if -7.1999999999999996e-65 < a

   1. Initial program 79.2%

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

      1. associate-*l/ N/A

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

      2. frac-sub N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      17. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

      2. PI-lowering-PI.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 72.4%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

         \[\leadsto {\left(\left(2 \cdot \left(b \cdot a\right)\right) \cdot b\right)}^{-1} \cdot \mathsf{PI}\left(\right) \]

      5. *-commutative N/A

         \[\leadsto {\left(b \cdot \left(2 \cdot \left(b \cdot a\right)\right)\right)}^{-1} \cdot \mathsf{PI}\left(\right) \]

      6. pow-prod-down N/A

         \[\leadsto \left({b}^{-1} \cdot {\left(2 \cdot \left(b \cdot a\right)\right)}^{-1}\right) \cdot \mathsf{PI}\left(\right) \]

      7. inv-pow N/A

         \[\leadsto \left(\frac{1}{b} \cdot {\left(2 \cdot \left(b \cdot a\right)\right)}^{-1}\right) \cdot \mathsf{PI}\left(\right) \]

      8. inv-pow N/A

         \[\leadsto \left(\frac{1}{b} \cdot \frac{1}{2 \cdot \left(b \cdot a\right)}\right) \cdot \mathsf{PI}\left(\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{1}{2 \cdot \left(b \cdot a\right)}\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{1}{2 \cdot \left(a \cdot b\right)}\right), \mathsf{PI}\left(\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{\frac{1}{2}}{a \cdot b}\right), \mathsf{PI}\left(\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{b} \cdot \frac{\frac{1}{2}}{a \cdot b}\right), \mathsf{PI}\left(\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot \frac{\frac{1}{2}}{a \cdot b}}{b}\right), \mathsf{PI}\left(\right)\right) \]

      14. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{1}{2}}{a \cdot b}}{b}\right), \mathsf{PI}\left(\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{a \cdot b}\right), b\right), \mathsf{PI}\left(\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot b\right)\right), b\right), \mathsf{PI}\left(\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), b\right), \mathsf{PI}\left(\right)\right) \]

      18. PI-lowering-PI.f64 72.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), b\right), \mathsf{PI.f64}\left(\right)\right) \]

   9. Applied egg-rr 72.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot b}\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5}{a} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (* a b))))
   (if (<= a -7.2e-65) (* (/ 0.5 a) t_0) (* t_0 (/ 0.5 b)))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) / (a * b);
	double tmp;
	if (a <= -7.2e-65) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI / (a * b);
	double tmp;
	if (a <= -7.2e-65) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi / (a * b)
	tmp = 0
	if a <= -7.2e-65:
		tmp = (0.5 / a) * t_0
	else:
		tmp = t_0 * (0.5 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi / Float64(a * b))
	tmp = 0.0
	if (a <= -7.2e-65)
		tmp = Float64(Float64(0.5 / a) * t_0);
	else
		tmp = Float64(t_0 * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi / (a * b);
	tmp = 0.0;
	if (a <= -7.2e-65)
		tmp = (0.5 / a) * t_0;
	else
		tmp = t_0 * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.2e-65], N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.1999999999999996e-65

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left({a}^{2} \cdot b\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right), \left(\color{blue}{{a}^{2}} \cdot b\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \left({a}^{\color{blue}{2}} \cdot b\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{b}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(a \cdot a\right), b\right)\right) \]

      7. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right) \]

   5. Simplified 78.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b \cdot a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{b \cdot a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot a\right)}\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} \cdot a\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot \color{blue}{b}\right)\right)\right) \]

      9. *-lowering-*.f64 90.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   7. Applied egg-rr 90.2%

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

   if -7.1999999999999996e-65 < a

   1. Initial program 79.2%

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

      1. associate-*l/ N/A

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

      2. frac-sub N/A

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

      3. frac-times N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

      17. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

   4. Applied egg-rr 79.7%

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

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

      2. PI-lowering-PI.f64 72.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 72.4%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{1}{2 \cdot b} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot b}\right), \color{blue}{\left(\frac{1}{2 \cdot b}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot b\right)\right), \left(\frac{\color{blue}{1}}{2 \cdot b}\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot b\right)\right), \left(\frac{1}{2 \cdot b}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{1}{2 \cdot b}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{1}{b \cdot \color{blue}{2}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{1}{b \cdot \frac{1}{\color{blue}{\frac{1}{2}}}}\right)\right) \]

      14. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{1}{\frac{b}{\color{blue}{\frac{1}{2}}}}\right)\right) \]

      15. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{\frac{1}{2}}{\color{blue}{b}}\right)\right) \]

      16. /-lowering-/.f64 72.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 72.4%

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

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. associate-*l/ N/A

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

   2. frac-sub N/A

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

   3. frac-times N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right), \color{blue}{\left(2 \cdot \left(a \cdot b\right)\right)}\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a \cdot 1\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(\color{blue}{2} \cdot \left(a \cdot b\right)\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(b - a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \left(2 \cdot \left(a \cdot b\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \left(b \cdot \color{blue}{a}\right)\right)\right) \]

   17. *-lowering-*.f64 80.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{a}\right)\right)\right) \]

4. Applied egg-rr 80.1%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot \left(b \cdot a\right)}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a} \cdot \left(b - a\right)\right)}\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot a}\right), \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot a}\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{b \cdot b - a \cdot a} \cdot \left(b - a\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot a\right)\right), \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{b \cdot b - a \cdot a}} \cdot \left(b - a\right)\right)\right) \]

   9. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(b - a\right)\right)}\right)\right) \]

   11. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{b - a}}}\right)\right) \]

   12. flip-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{b + \color{blue}{a}}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\frac{1}{b + a}\right)}\right)\right) \]

   14. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\color{blue}{1}}{b + a}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(1, \color{blue}{\left(b + a\right)}\right)\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(1, \left(a + \color{blue}{b}\right)\right)\right)\right) \]

   17. +-lowering-+.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a, \color{blue}{b}\right)\right)\right)\right) \]

6. Applied egg-rr 99.6%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. frac-times N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \color{blue}{\left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \left(\left(a + \color{blue}{b}\right) \cdot \left(a \cdot b\right)\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(\color{blue}{\left(a + b\right)} \cdot \left(a \cdot b\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(\color{blue}{\left(a + b\right)} \cdot \left(a \cdot b\right)\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\color{blue}{a} + b\right) \cdot \left(a \cdot b\right)\right)\right) \]

   9. remove-double-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{1}{\frac{1}{a + b}} \cdot \left(\color{blue}{a} \cdot b\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{1 \cdot \left(a \cdot b\right)}{\color{blue}{\frac{1}{a + b}}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)}{\frac{1}{a + b}}\right)\right) \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)}{\frac{\color{blue}{1}}{a + b}}\right)\right) \]

   13. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)}{\frac{1}{a + b}}\right)\right) \]

   14. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\frac{a \cdot b}{\frac{\color{blue}{1}}{a + b}}\right)\right) \]

   15. un-div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(a \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{a + b}}}\right)\right) \]

   16. remove-double-div N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(a \cdot b\right) \cdot \left(a + \color{blue}{b}\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\left(a \cdot b\right), \color{blue}{\left(a + b\right)}\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\color{blue}{a} + b\right)\right)\right) \]

   19. +-lowering-+.f64 99.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{+.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{\left(a \cdot b\right) \cdot \left(a + b\right)}} \]
9. Add Preprocessing

Alternative 9: 62.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left({a}^{2} \cdot b\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right), \left(\color{blue}{{a}^{2}} \cdot b\right)\right) \]

   4. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \left({a}^{\color{blue}{2}} \cdot b\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{b}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(a \cdot a\right), b\right)\right) \]

   7. *-lowering-*.f64 56.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right) \]

5. Simplified 56.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{b \cdot a}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{b \cdot a}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot a\right)}\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} \cdot a\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot \color{blue}{b}\right)\right)\right) \]

   9. *-lowering-*.f64 62.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

7. Applied egg-rr 62.8%

   \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(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))))