NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 10.1s

Alternatives: 10

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: 78.4% 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: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(Pi / N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b * a), $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. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-rgt-identity N/A

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

   12. *-lft-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.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 rewrites 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. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

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

Alternative 2: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e+121)
   (/ (/ (* PI 0.5) (* b a)) a)
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6e+121) {
		tmp = ((((double) M_PI) * 0.5) / (b * a)) / 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 <= -8.6e+121) {
		tmp = ((Math.PI * 0.5) / (b * a)) / a;
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.6e+121:
		tmp = ((math.pi * 0.5) / (b * a)) / a
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6e+121)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / a);
	else
		tmp = Float64(Float64(pi * 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 <= -8.6e+121)
		tmp = ((pi * 0.5) / (b * a)) / a;
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6e+121], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5999999999999994e121

   1. Initial program 71.8%

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

   3. Taylor expanded in b around 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 99.9%

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

   if -8.5999999999999994e121 < a

   1. Initial program 83.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. times-frac N/A

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

      14. lower-*.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 rewrites 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. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-times N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 94.1%

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

Alternative 3: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e+121)
   (/ (* PI (/ 0.5 (* b a))) a)
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6e+121) {
		tmp = (((double) M_PI) * (0.5 / (b * a))) / 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 <= -8.6e+121) {
		tmp = (Math.PI * (0.5 / (b * a))) / a;
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.6e+121:
		tmp = (math.pi * (0.5 / (b * a))) / a
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6e+121)
		tmp = Float64(Float64(pi * Float64(0.5 / Float64(b * a))) / a);
	else
		tmp = Float64(Float64(pi * 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 <= -8.6e+121)
		tmp = (pi * (0.5 / (b * a))) / a;
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6e+121], N[(N[(Pi * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5999999999999994e121

   1. Initial program 71.8%

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

   3. Taylor expanded in b around 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 85.6%

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

   9. Applied rewrites 99.9%

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

   if -8.5999999999999994e121 < a

   1. Initial program 83.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. times-frac N/A

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

      14. lower-*.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 rewrites 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. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-times N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 94.1%

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

Alternative 4: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e+121)
   (* (/ 0.5 (* b a)) (/ PI a))
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6e+121) {
		tmp = (0.5 / (b * a)) * (((double) M_PI) / 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 <= -8.6e+121) {
		tmp = (0.5 / (b * a)) * (Math.PI / a);
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.6e+121:
		tmp = (0.5 / (b * a)) * (math.pi / a)
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6e+121)
		tmp = Float64(Float64(0.5 / Float64(b * a)) * Float64(pi / a));
	else
		tmp = Float64(Float64(pi * 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 <= -8.6e+121)
		tmp = (0.5 / (b * a)) * (pi / a);
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6e+121], N[(N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5999999999999994e121

   1. Initial program 71.8%

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

   3. Taylor expanded in b around 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 99.8%

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

   if -8.5999999999999994e121 < a

   1. Initial program 83.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. times-frac N/A

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

      14. lower-*.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 rewrites 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. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-times N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 94.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+43}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e+43)
   (/ (* PI 0.5) (* a (* b (+ b a))))
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e+43) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * (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 (b <= 1e+43) {
		tmp = (Math.PI * 0.5) / (a * (b * (b + a)));
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e+43:
		tmp = (math.pi * 0.5) / (a * (b * (b + a)))
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e+43)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * Float64(b + a))));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e+43)
		tmp = (pi * 0.5) / (a * (b * (b + a)));
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e+43], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.00000000000000001e43

   1. Initial program 81.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. times-frac N/A

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

      14. lower-*.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 rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-times N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. un-div-inv N/A

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

   10. Applied rewrites 96.6%

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

   if 1.00000000000000001e43 < b

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. times-frac N/A

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

      14. lower-*.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 rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-times N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+43}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b + a), $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. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-rgt-identity N/A

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

   12. *-lft-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.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 rewrites 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. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f64 N/A

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

   7. unpow-prod-down N/A

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

   8. inv-pow N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. inv-pow N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-/r* N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 7: 71.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e-51) (/ (* PI 0.5) (* a (* b a))) (/ (* PI 0.5) (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.5e-51) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * 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 <= 2.5e-51) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (Math.PI * 0.5) / (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.5e-51:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (math.pi * 0.5) / (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.5e-51)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	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 <= 2.5e-51)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (pi * 0.5) / (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.5e-51], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $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 < 2.50000000000000002e-51

   1. Initial program 79.5%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 2.50000000000000002e-51 < b

   1. Initial program 87.2%

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

   3. 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.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. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-rgt-identity N/A

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

   12. *-lft-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.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 rewrites 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. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. un-div-inv N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. frac-times N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. times-frac N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. frac-times N/A

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-/.f64 N/A

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

   19. un-div-inv N/A

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

10. Applied rewrites 95.3%

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

11. Final simplification 95.3%

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

Alternative 9: 62.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $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 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 63.6

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

5. Applied rewrites 63.6%

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

6. Final simplification 63.6%

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

Alternative 10: 56.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(Pi * N[(0.5 / N[(b * N[(a * a), $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 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 63.6

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

5. Applied rewrites 63.6%

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

7. Applied rewrites 57.8%

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

Reproduce

herbie shell --seed 2024232 
(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))))