NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 10.6s

Alternatives: 11

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. associate-*r/ N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. *-lft-identity N/A

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

   8. times-frac N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in b around 0

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

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

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

   2. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 96.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot 2}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6e+82)
   (/ (* PI 0.5) (* a (* b (+ a b))))
   (/ (/ PI (* b 2.0)) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6e+82) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = (((double) M_PI) / (b * 2.0)) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6e+82) {
		tmp = (Math.PI * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = (Math.PI / (b * 2.0)) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6e+82:
		tmp = (math.pi * 0.5) / (a * (b * (a + b)))
	else:
		tmp = (math.pi / (b * 2.0)) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6e+82)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(pi / Float64(b * 2.0)) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6e+82)
		tmp = (pi * 0.5) / (a * (b * (a + b)));
	else
		tmp = (pi / (b * 2.0)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6e+82], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.99999999999999978e82

   1. Initial program 82.6%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-*r/ N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. times-frac N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 99.6

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

   7. Simplified 99.6%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 94.8

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

   9. Applied egg-rr 94.8%

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

   if 5.99999999999999978e82 < b

   1. Initial program 77.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 87.0

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

   5. Simplified 87.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot 2}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6e+82)
   (/ (* PI 0.5) (* a (* b (+ a b))))
   (* (/ PI (* a b)) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6e+82) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6e+82) {
		tmp = (Math.PI * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = (Math.PI / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6e+82:
		tmp = (math.pi * 0.5) / (a * (b * (a + b)))
	else:
		tmp = (math.pi / (a * b)) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6e+82)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6e+82)
		tmp = (pi * 0.5) / (a * (b * (a + b)));
	else
		tmp = (pi / (a * b)) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6e+82], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.99999999999999978e82

   1. Initial program 82.6%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-*r/ N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. times-frac N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 99.6

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

   7. Simplified 99.6%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 94.8

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

   9. Applied egg-rr 94.8%

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

   if 5.99999999999999978e82 < b

   1. Initial program 77.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 87.0

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

   5. Simplified 87.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. associate-*r/ N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. *-lft-identity N/A

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

   8. times-frac N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in b around 0

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

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

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

   2. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

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

   1. add-sqr-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. sqrt-prod N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f64 99.5

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

9. Applied egg-rr 99.5%

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

    1. associate-*l/ N/A

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

    2. frac-2neg N/A

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

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

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

11. Applied egg-rr 99.6%

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

Alternative 5: 95.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6e+82)
   (/ (* PI 0.5) (* a (* b (+ a b))))
   (/ (* PI 0.5) (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6e+82) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6e+82) {
		tmp = (Math.PI * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6e+82:
		tmp = (math.pi * 0.5) / (a * (b * (a + b)))
	else:
		tmp = (math.pi * 0.5) / (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6e+82)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6e+82)
		tmp = (pi * 0.5) / (a * (b * (a + b)));
	else
		tmp = (pi * 0.5) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6e+82], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.99999999999999978e82

   1. Initial program 82.6%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-*r/ N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. times-frac N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 99.6

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

   7. Simplified 99.6%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 94.8

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

   9. Applied egg-rr 94.8%

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

   if 5.99999999999999978e82 < b

   1. Initial program 77.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 87.0

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

   5. Simplified 87.0%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.36e-28)
   (/ (* PI 0.5) (* a (* a b)))
   (/ (* PI 0.5) (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.36e-28) {
		tmp = (((double) M_PI) * 0.5) / (a * (a * b));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.36e-28) {
		tmp = (Math.PI * 0.5) / (a * (a * b));
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.36e-28:
		tmp = (math.pi * 0.5) / (a * (a * b))
	else:
		tmp = (math.pi * 0.5) / (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.36e-28)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.36e-28)
		tmp = (pi * 0.5) / (a * (a * b));
	else
		tmp = (pi * 0.5) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.36e-28], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35999999999999989e-28

   1. Initial program 79.9%

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 72.8

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

   5. Simplified 72.8%

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

   if 1.35999999999999989e-28 < b

   1. Initial program 86.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 88.9

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

   7. Applied egg-rr 88.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4e-28) (/ (* PI 0.5) (* a (* a b))) (/ (* PI 0.5) (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4e-28) {
		tmp = (((double) M_PI) * 0.5) / (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 (b <= 1.4e-28) {
		tmp = (Math.PI * 0.5) / (a * (a * b));
	} else {
		tmp = (Math.PI * 0.5) / (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.4e-28:
		tmp = (math.pi * 0.5) / (a * (a * b))
	else:
		tmp = (math.pi * 0.5) / (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4e-28)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.4e-28)
		tmp = (pi * 0.5) / (a * (a * b));
	else
		tmp = (pi * 0.5) / (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4e-28], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999e-28

   1. Initial program 79.9%

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 72.8

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

   5. Simplified 72.8%

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

   if 1.3999999999999999e-28 < b

   1. Initial program 86.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22e-28)
   (/ (* PI 0.5) (* a (* a b)))
   (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22e-28) {
		tmp = (((double) M_PI) * 0.5) / (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 (b <= 1.22e-28) {
		tmp = (Math.PI * 0.5) / (a * (a * b));
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.22e-28:
		tmp = (math.pi * 0.5) / (a * (a * b))
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.22e-28)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.22e-28)
		tmp = (pi * 0.5) / (a * (a * b));
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.22e-28], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22e-28

   1. Initial program 79.9%

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 72.8

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

   5. Simplified 72.8%

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

   if 1.22e-28 < b

   1. Initial program 86.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 81.3

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

   7. Applied egg-rr 81.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-63}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e-63) (* PI (/ 0.5 (* b (* a a)))) (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-63) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-63) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.3e-63:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.3e-63)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.3e-63)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.3e-63], N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.3e-63

   1. Initial program 79.6%

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

   3. 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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 73.2

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

   5. Simplified 73.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 66.9

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

   7. Applied egg-rr 66.9%

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

   if 2.3e-63 < b

   1. Initial program 86.6%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 78.2

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

   5. Simplified 78.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.3

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

   7. Applied egg-rr 78.3%

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

Alternative 10: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. associate-*r/ N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. *-lft-identity N/A

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

   8. times-frac N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in b around 0

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

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

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

   2. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

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

   1. add-sqr-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. sqrt-prod N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f64 99.5

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

9. Applied egg-rr 99.5%

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

    1. frac-times N/A

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

    2. frac-2neg N/A

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

    3. *-lft-identity N/A

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

    4. sqrt-unprod N/A

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

    5. associate-*r* N/A

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

    6. add-sqr-sqrt N/A

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

    7. sqrt-unprod N/A

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

    8. add-sqr-sqrt N/A

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

    9. associate-*r* N/A

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

    10. +-commutative N/A

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

11. Applied egg-rr 99.2%

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

12. Final simplification 99.2%

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

Alternative 11: 56.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 56.7

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

5. Simplified 56.7%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 56.7

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

7. Applied egg-rr 56.7%

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

Reproduce

herbie shell --seed 2024204 
(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))))