NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 11.0s

Alternatives: 9

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, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. *-rgt-identity N/A

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

   6. *-lft-identity N/A

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

   7. times-frac N/A

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

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

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

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. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. div-inv N/A

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

   4. difference-of-squares N/A

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

   5. times-frac N/A

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.6%

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

Alternative 3: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+146)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+146) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+146) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+146:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = math.pi * (0.5 / (b * (a * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+146)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+146)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = pi * (0.5 / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+146], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.99999999999999934e145

   1. Initial program 51.7%

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

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

      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 97.5

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

   5. Simplified 97.5%

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

   if -9.99999999999999934e145 < a

   1. Initial program 80.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. times-frac N/A

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

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

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

   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. Step-by-step derivation

      1. clear-num N/A

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

      2. frac-times N/A

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

      3. associate-*l* N/A

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

      4. *-un-lft-identity N/A

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

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

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

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

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

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

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

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

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

      9. div-inv N/A

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

      10. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      16. --lowering--.f64 99.2

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

   6. Applied egg-rr 99.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. difference-of-squares N/A

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

      8. associate-*r/ N/A

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

      9. flip-+ N/A

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

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

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

   8. Applied egg-rr 95.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e-122)
   (/ (* PI 0.5) (* a (* b a)))
   (/ (* PI 0.5) (* b (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-122) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (b * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-122) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (Math.PI * 0.5) / (b * (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.2e-122:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (math.pi * 0.5) / (b * (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e-122)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.2e-122)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (pi * 0.5) / (b * (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e-122], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.19999999999999994e-122

   1. Initial program 77.5%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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 82.2

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

   5. Simplified 82.2%

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

   if -1.19999999999999994e-122 < a

   1. Initial program 76.7%

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

      1. *-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. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. times-frac N/A

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

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

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

   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 inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 62.2

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

   7. Simplified 62.2%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. *-lowering-*.f64 71.5

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

   9. Applied egg-rr 71.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e-122)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-122) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-122) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.2e-122:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e-122)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.2e-122)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e-122], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.19999999999999994e-122

   1. Initial program 77.5%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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 82.2

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

   5. Simplified 82.2%

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

   if -1.19999999999999994e-122 < a

   1. Initial program 76.7%

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

      1. *-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. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. times-frac N/A

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

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

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

   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 inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 62.2

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

   7. Simplified 62.2%

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. PI-lowering-PI.f64 71.4

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

   9. Applied egg-rr 71.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e-122)
   (* PI (/ 0.5 (* b (* a a))))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-122) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-122) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.2e-122:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e-122)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.2e-122)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e-122], N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.19999999999999994e-122

   1. Initial program 77.5%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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 82.2

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

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}} \]
   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 75.4

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

   7. Applied egg-rr 75.4%

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

   if -1.19999999999999994e-122 < a

   1. Initial program 76.7%

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

      1. *-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. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. times-frac N/A

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

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

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

   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 inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 62.2

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

   7. Simplified 62.2%

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. PI-lowering-PI.f64 71.4

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

   9. Applied egg-rr 71.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e-122)
   (* PI (/ 0.5 (* b (* a a))))
   (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e-122) {
		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 (a <= -1.2e-122) {
		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 a <= -1.2e-122:
		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 (a <= -1.2e-122)
		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 (a <= -1.2e-122)
		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[a, -1.2e-122], 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 a < -1.19999999999999994e-122

   1. Initial program 77.5%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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 82.2

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

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}} \]
   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 75.4

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

   7. Applied egg-rr 75.4%

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

   if -1.19999999999999994e-122 < a

   1. Initial program 76.7%

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

      1. *-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. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. times-frac N/A

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

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

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

   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 inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 62.2

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

   7. Simplified 62.2%

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

Alternative 8: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. *-rgt-identity N/A

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

   6. *-lft-identity N/A

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

   7. times-frac N/A

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

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

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

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. Step-by-step derivation

   1. clear-num N/A

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

   2. frac-times N/A

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

   3. associate-*l* N/A

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

   4. *-un-lft-identity N/A

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

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

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

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

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

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

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

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

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

   9. div-inv N/A

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

   10. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   16. --lowering--.f64 99.0

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

6. Applied egg-rr 99.0%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. difference-of-squares N/A

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

   6. associate-*r/ N/A

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

   7. flip-+ N/A

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

   8. *-commutative N/A

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

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

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

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

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

   11. *-lowering-*.f64 99.0

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 9: 57.0% 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 77.0%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. *-rgt-identity N/A

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

   6. *-lft-identity N/A

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

   7. times-frac N/A

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

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

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

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 inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
6. 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. *-commutative N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 60.0

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

7. Simplified 60.0%

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

Reproduce

herbie shell --seed 2024199 
(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))))