NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 3.3s

Alternatives: 6

Speedup: 1.8×

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: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a + b\right) \cdot 2\\ \mathbf{if}\;a \leq -1 \cdot 10^{+130}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-1}{b}}{t\_0}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{\pi}{t\_0 \cdot \left(-a\right)} \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (+ a b) 2.0)))
   (if (<= a -1e+130)
     (* (/ PI (- b a)) (/ (/ -1.0 b) t_0))
     (if (<= a 5e+147)
       (* (/ PI (* t_0 (- a))) (/ -1.0 b))
       (* (/ (/ PI a) (* b a)) 0.5)))))

C

double code(double a, double b) {
	double t_0 = (a + b) * 2.0;
	double tmp;
	if (a <= -1e+130) {
		tmp = (((double) M_PI) / (b - a)) * ((-1.0 / b) / t_0);
	} else if (a <= 5e+147) {
		tmp = (((double) M_PI) / (t_0 * -a)) * (-1.0 / b);
	} else {
		tmp = ((((double) M_PI) / a) / (b * a)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (a + b) * 2.0;
	double tmp;
	if (a <= -1e+130) {
		tmp = (Math.PI / (b - a)) * ((-1.0 / b) / t_0);
	} else if (a <= 5e+147) {
		tmp = (Math.PI / (t_0 * -a)) * (-1.0 / b);
	} else {
		tmp = ((Math.PI / a) / (b * a)) * 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a + b) * 2.0
	tmp = 0
	if a <= -1e+130:
		tmp = (math.pi / (b - a)) * ((-1.0 / b) / t_0)
	elif a <= 5e+147:
		tmp = (math.pi / (t_0 * -a)) * (-1.0 / b)
	else:
		tmp = ((math.pi / a) / (b * a)) * 0.5
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a + b) * 2.0)
	tmp = 0.0
	if (a <= -1e+130)
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(Float64(-1.0 / b) / t_0));
	elseif (a <= 5e+147)
		tmp = Float64(Float64(pi / Float64(t_0 * Float64(-a))) * Float64(-1.0 / b));
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a + b) * 2.0;
	tmp = 0.0;
	if (a <= -1e+130)
		tmp = (pi / (b - a)) * ((-1.0 / b) / t_0);
	elseif (a <= 5e+147)
		tmp = (pi / (t_0 * -a)) * (-1.0 / b);
	else
		tmp = ((pi / a) / (b * a)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[a, -1e+130], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / b), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e+147], N[(N[(Pi / N[(t$95$0 * (-a)), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.0000000000000001e130

   1. Initial program 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 80.2

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

   3. Applied rewrites 80.2%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 80.2

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

   6. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. associate-*r* N/A

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

   8. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8

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

   10. Applied rewrites 99.8%

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

   if -1.0000000000000001e130 < a < 5.0000000000000002e147

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 91.7

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

   3. Applied rewrites 91.7%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.6

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

   9. Applied rewrites 99.6%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-rgt-identity N/A

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

       4. lift-PI.f64 99.6

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

       5. lift-*.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. +-commutative N/A

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

       13. lower-+.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 5.0000000000000002e147 < a

   1. Initial program 50.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

Alternative 2: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{if}\;a \leq -9.1 \cdot 10^{+167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{\pi}{\left(\left(a + b\right) \cdot 2\right) \cdot \left(-a\right)} \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI a) (* b a)) 0.5)))
   (if (<= a -9.1e+167)
     t_0
     (if (<= a 5e+147) (* (/ PI (* (* (+ a b) 2.0) (- a))) (/ -1.0 b)) t_0))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -9.1e+167) {
		tmp = t_0;
	} else if (a <= 5e+147) {
		tmp = (((double) M_PI) / (((a + b) * 2.0) * -a)) * (-1.0 / b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -9.1e+167) {
		tmp = t_0;
	} else if (a <= 5e+147) {
		tmp = (Math.PI / (((a + b) * 2.0) * -a)) * (-1.0 / b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / a) / (b * a)) * 0.5
	tmp = 0
	if a <= -9.1e+167:
		tmp = t_0
	elif a <= 5e+147:
		tmp = (math.pi / (((a + b) * 2.0) * -a)) * (-1.0 / b)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5)
	tmp = 0.0
	if (a <= -9.1e+167)
		tmp = t_0;
	elseif (a <= 5e+147)
		tmp = Float64(Float64(pi / Float64(Float64(Float64(a + b) * 2.0) * Float64(-a))) * Float64(-1.0 / b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / a) / (b * a)) * 0.5;
	tmp = 0.0;
	if (a <= -9.1e+167)
		tmp = t_0;
	elseif (a <= 5e+147)
		tmp = (pi / (((a + b) * 2.0) * -a)) * (-1.0 / b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -9.1e+167], t$95$0, If[LessEqual[a, 5e+147], N[(N[(Pi / N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.10000000000000019e167 or 5.0000000000000002e147 < a

   1. Initial program 51.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.8

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

   4. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if -9.10000000000000019e167 < a < 5.0000000000000002e147

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 91.3

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

   3. Applied rewrites 91.3%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 59.1

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

   6. Applied rewrites 59.1%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.8

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

   9. Applied rewrites 98.8%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-rgt-identity N/A

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

       4. lift-PI.f64 98.8

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

       5. lift-*.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. +-commutative N/A

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

       13. lower-+.f64 98.8

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

   11. Applied rewrites 98.8%

       \[\leadsto \color{blue}{\frac{\pi}{\left(\left(a + b\right) \cdot 2\right) \cdot \left(-a\right)}} \cdot \frac{-1}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* b (* b a))) 0.5)))
   (if (<= b -2.5e-17)
     t_0
     (if (<= b 2.95e-5) (* (/ (/ PI a) (* b a)) 0.5) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / (b * (b * a))) * 0.5;
	double tmp;
	if (b <= -2.5e-17) {
		tmp = t_0;
	} else if (b <= 2.95e-5) {
		tmp = ((((double) M_PI) / a) / (b * a)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / (b * (b * a))) * 0.5;
	double tmp;
	if (b <= -2.5e-17) {
		tmp = t_0;
	} else if (b <= 2.95e-5) {
		tmp = ((Math.PI / a) / (b * a)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / (b * (b * a))) * 0.5
	tmp = 0
	if b <= -2.5e-17:
		tmp = t_0
	elif b <= 2.95e-5:
		tmp = ((math.pi / a) / (b * a)) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(b * Float64(b * a))) * 0.5)
	tmp = 0.0
	if (b <= -2.5e-17)
		tmp = t_0;
	elseif (b <= 2.95e-5)
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / (b * (b * a))) * 0.5;
	tmp = 0.0;
	if (b <= -2.5e-17)
		tmp = t_0;
	elseif (b <= 2.95e-5)
		tmp = ((pi / a) / (b * a)) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, -2.5e-17], t$95$0, If[LessEqual[b, 2.95e-5], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4999999999999999e-17 or 2.9499999999999999e-5 < b

   1. Initial program 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-PI.f64 88.6

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 79.5

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

   8. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.9

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

   10. Applied rewrites 89.9%

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

   if -2.4999999999999999e-17 < b < 2.9499999999999999e-5

   1. Initial program 80.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

Alternative 4: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* b (* b a))) 0.5)))
   (if (<= b -2.5e-17)
     t_0
     (if (<= b 2.95e-5) (* (/ PI (* a (* a b))) 0.5) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / (b * (b * a))) * 0.5;
	double tmp;
	if (b <= -2.5e-17) {
		tmp = t_0;
	} else if (b <= 2.95e-5) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / (b * (b * a))) * 0.5;
	double tmp;
	if (b <= -2.5e-17) {
		tmp = t_0;
	} else if (b <= 2.95e-5) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / (b * (b * a))) * 0.5
	tmp = 0
	if b <= -2.5e-17:
		tmp = t_0
	elif b <= 2.95e-5:
		tmp = (math.pi / (a * (a * b))) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(b * Float64(b * a))) * 0.5)
	tmp = 0.0
	if (b <= -2.5e-17)
		tmp = t_0;
	elseif (b <= 2.95e-5)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / (b * (b * a))) * 0.5;
	tmp = 0.0;
	if (b <= -2.5e-17)
		tmp = t_0;
	elseif (b <= 2.95e-5)
		tmp = (pi / (a * (a * b))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, -2.5e-17], t$95$0, If[LessEqual[b, 2.95e-5], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4999999999999999e-17 or 2.9499999999999999e-5 < b

   1. Initial program 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-PI.f64 88.6

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 79.5

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

   8. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.9

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

   10. Applied rewrites 89.9%

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

   if -2.4999999999999999e-17 < b < 2.9499999999999999e-5

   1. Initial program 80.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

Alternative 5: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* (* b b) a)) 0.5)))
   (if (<= b -2.5e-17)
     t_0
     (if (<= b 2.95e-5) (* (/ PI (* a (* a b))) 0.5) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / ((b * b) * a)) * 0.5;
	double tmp;
	if (b <= -2.5e-17) {
		tmp = t_0;
	} else if (b <= 2.95e-5) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / ((b * b) * a)) * 0.5;
	double tmp;
	if (b <= -2.5e-17) {
		tmp = t_0;
	} else if (b <= 2.95e-5) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / ((b * b) * a)) * 0.5
	tmp = 0
	if b <= -2.5e-17:
		tmp = t_0
	elif b <= 2.95e-5:
		tmp = (math.pi / (a * (a * b))) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5)
	tmp = 0.0
	if (b <= -2.5e-17)
		tmp = t_0;
	elseif (b <= 2.95e-5)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / ((b * b) * a)) * 0.5;
	tmp = 0.0;
	if (b <= -2.5e-17)
		tmp = t_0;
	elseif (b <= 2.95e-5)
		tmp = (pi / (a * (a * b))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, -2.5e-17], t$95$0, If[LessEqual[b, 2.95e-5], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4999999999999999e-17 or 2.9499999999999999e-5 < b

   1. Initial program 76.4%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if -2.4999999999999999e-17 < b < 2.9499999999999999e-5

   1. Initial program 80.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

Alternative 6: 62.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 56.7

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

4. Applied rewrites 56.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 62.7

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

6. Applied rewrites 62.7%

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

Reproduce

herbie shell --seed 2025101 
(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))))