NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 96.5%

Time: 3.6s

Alternatives: 9

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: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{b - a} \cdot \left(b - a\right)}{a \cdot b}\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(a \cdot b\right) \cdot b}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+131}:\\ \;\;\;\;\frac{\pi \cdot 1}{2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) (* (* a b) 2.0))))
   (if (<= a -6.6e+154)
     t_0
     (if (<= a -6e-211)
       (/ (* (/ (/ PI (* (+ a b) 2.0)) (- b a)) (- b a)) (* a b))
       (if (<= a 3.3e-174)
         (/ (* 0.5 PI) (* (* a b) b))
         (if (<= a 1.15e+131)
           (*
            (/ (* PI 1.0) (* 2.0 (* (+ b a) (- b a))))
            (- (/ 1.0 a) (/ 1.0 b)))
           t_0))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -6.6e+154) {
		tmp = t_0;
	} else if (a <= -6e-211) {
		tmp = (((((double) M_PI) / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b);
	} else if (a <= 3.3e-174) {
		tmp = (0.5 * ((double) M_PI)) / ((a * b) * b);
	} else if (a <= 1.15e+131) {
		tmp = ((((double) M_PI) * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / 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) / ((a * b) * 2.0);
	double tmp;
	if (a <= -6.6e+154) {
		tmp = t_0;
	} else if (a <= -6e-211) {
		tmp = (((Math.PI / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b);
	} else if (a <= 3.3e-174) {
		tmp = (0.5 * Math.PI) / ((a * b) * b);
	} else if (a <= 1.15e+131) {
		tmp = ((Math.PI * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / ((a * b) * 2.0)
	tmp = 0
	if a <= -6.6e+154:
		tmp = t_0
	elif a <= -6e-211:
		tmp = (((math.pi / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b)
	elif a <= 3.3e-174:
		tmp = (0.5 * math.pi) / ((a * b) * b)
	elif a <= 1.15e+131:
		tmp = ((math.pi * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / a) - (1.0 / b))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0))
	tmp = 0.0
	if (a <= -6.6e+154)
		tmp = t_0;
	elseif (a <= -6e-211)
		tmp = Float64(Float64(Float64(Float64(pi / Float64(Float64(a + b) * 2.0)) / Float64(b - a)) * Float64(b - a)) / Float64(a * b));
	elseif (a <= 3.3e-174)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(a * b) * b));
	elseif (a <= 1.15e+131)
		tmp = Float64(Float64(Float64(pi * 1.0) / Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / ((a * b) * 2.0);
	tmp = 0.0;
	if (a <= -6.6e+154)
		tmp = t_0;
	elseif (a <= -6e-211)
		tmp = (((pi / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b);
	elseif (a <= 3.3e-174)
		tmp = (0.5 * pi) / ((a * b) * b);
	elseif (a <= 1.15e+131)
		tmp = ((pi * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / a) - (1.0 / b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.6e+154], t$95$0, If[LessEqual[a, -6e-211], N[(N[(N[(N[(Pi / N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.3e-174], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.15e+131], N[(N[(N[(Pi * 1.0), $MachinePrecision] / N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.6e154 or 1.14999999999999996e131 < a

   1. Initial program 53.8%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 78.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -6.6e154 < a < -6.00000000000000009e-211

   1. Initial program 92.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 95.2

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

   3. Applied rewrites 95.2%

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

   5. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 95.4%

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

   if -6.00000000000000009e-211 < a < 3.3000000000000001e-174

   1. Initial program 71.4%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 80.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 75.9

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

   6. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 76.0

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 3.3000000000000001e-174 < a < 1.14999999999999996e131

   1. Initial program 94.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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 95.7

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

   3. Applied rewrites 95.7%

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

Alternative 2: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{b - a} \cdot \left(b - a\right)}{a \cdot b}\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(a \cdot b\right) \cdot b}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{\pi}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \cdot \frac{b - a}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) (* (* a b) 2.0))))
   (if (<= a -6.6e+154)
     t_0
     (if (<= a -6e-211)
       (/ (* (/ (/ PI (* (+ a b) 2.0)) (- b a)) (- b a)) (* a b))
       (if (<= a 3.3e-174)
         (/ (* 0.5 PI) (* (* a b) b))
         (if (<= a 1.1e+131)
           (* (/ PI (* (* 2.0 (+ b a)) (- b a))) (/ (- b a) (* a b)))
           t_0))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -6.6e+154) {
		tmp = t_0;
	} else if (a <= -6e-211) {
		tmp = (((((double) M_PI) / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b);
	} else if (a <= 3.3e-174) {
		tmp = (0.5 * ((double) M_PI)) / ((a * b) * b);
	} else if (a <= 1.1e+131) {
		tmp = (((double) M_PI) / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -6.6e+154) {
		tmp = t_0;
	} else if (a <= -6e-211) {
		tmp = (((Math.PI / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b);
	} else if (a <= 3.3e-174) {
		tmp = (0.5 * Math.PI) / ((a * b) * b);
	} else if (a <= 1.1e+131) {
		tmp = (Math.PI / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / ((a * b) * 2.0)
	tmp = 0
	if a <= -6.6e+154:
		tmp = t_0
	elif a <= -6e-211:
		tmp = (((math.pi / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b)
	elif a <= 3.3e-174:
		tmp = (0.5 * math.pi) / ((a * b) * b)
	elif a <= 1.1e+131:
		tmp = (math.pi / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0))
	tmp = 0.0
	if (a <= -6.6e+154)
		tmp = t_0;
	elseif (a <= -6e-211)
		tmp = Float64(Float64(Float64(Float64(pi / Float64(Float64(a + b) * 2.0)) / Float64(b - a)) * Float64(b - a)) / Float64(a * b));
	elseif (a <= 3.3e-174)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(a * b) * b));
	elseif (a <= 1.1e+131)
		tmp = Float64(Float64(pi / Float64(Float64(2.0 * Float64(b + a)) * Float64(b - a))) * Float64(Float64(b - a) / Float64(a * b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / ((a * b) * 2.0);
	tmp = 0.0;
	if (a <= -6.6e+154)
		tmp = t_0;
	elseif (a <= -6e-211)
		tmp = (((pi / ((a + b) * 2.0)) / (b - a)) * (b - a)) / (a * b);
	elseif (a <= 3.3e-174)
		tmp = (0.5 * pi) / ((a * b) * b);
	elseif (a <= 1.1e+131)
		tmp = (pi / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.6e+154], t$95$0, If[LessEqual[a, -6e-211], N[(N[(N[(N[(Pi / N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.3e-174], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.1e+131], N[(N[(Pi / N[(N[(2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.6e154 or 1.0999999999999999e131 < a

   1. Initial program 53.8%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 78.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -6.6e154 < a < -6.00000000000000009e-211

   1. Initial program 92.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 95.2

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

   3. Applied rewrites 95.2%

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

   5. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 95.4%

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

   if -6.00000000000000009e-211 < a < 3.3000000000000001e-174

   1. Initial program 71.4%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 80.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 75.9

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

   6. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 76.0

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 3.3000000000000001e-174 < a < 1.0999999999999999e131

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 95.9

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

   3. Applied rewrites 95.9%

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

   5. Applied rewrites 95.7%

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

Alternative 3: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \cdot \frac{b - a}{a \cdot b}\\ t_1 := \frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{if}\;a \leq -5 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{-205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b \cdot a}}{b}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* (* 2.0 (+ b a)) (- b a))) (/ (- b a) (* a b))))
        (t_1 (/ (/ PI a) (* (* a b) 2.0))))
   (if (<= a -5e+107)
     t_1
     (if (<= a -7.5e-205)
       t_0
       (if (<= a 1.3e-173)
         (/ (/ (* 0.5 PI) (* b a)) b)
         (if (<= a 1.1e+131) t_0 t_1))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	double t_1 = (((double) M_PI) / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -5e+107) {
		tmp = t_1;
	} else if (a <= -7.5e-205) {
		tmp = t_0;
	} else if (a <= 1.3e-173) {
		tmp = ((0.5 * ((double) M_PI)) / (b * a)) / b;
	} else if (a <= 1.1e+131) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	double t_1 = (Math.PI / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -5e+107) {
		tmp = t_1;
	} else if (a <= -7.5e-205) {
		tmp = t_0;
	} else if (a <= 1.3e-173) {
		tmp = ((0.5 * Math.PI) / (b * a)) / b;
	} else if (a <= 1.1e+131) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b))
	t_1 = (math.pi / a) / ((a * b) * 2.0)
	tmp = 0
	if a <= -5e+107:
		tmp = t_1
	elif a <= -7.5e-205:
		tmp = t_0
	elif a <= 1.3e-173:
		tmp = ((0.5 * math.pi) / (b * a)) / b
	elif a <= 1.1e+131:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(Float64(2.0 * Float64(b + a)) * Float64(b - a))) * Float64(Float64(b - a) / Float64(a * b)))
	t_1 = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0))
	tmp = 0.0
	if (a <= -5e+107)
		tmp = t_1;
	elseif (a <= -7.5e-205)
		tmp = t_0;
	elseif (a <= 1.3e-173)
		tmp = Float64(Float64(Float64(0.5 * pi) / Float64(b * a)) / b);
	elseif (a <= 1.1e+131)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	t_1 = (pi / a) / ((a * b) * 2.0);
	tmp = 0.0;
	if (a <= -5e+107)
		tmp = t_1;
	elseif (a <= -7.5e-205)
		tmp = t_0;
	elseif (a <= 1.3e-173)
		tmp = ((0.5 * pi) / (b * a)) / b;
	elseif (a <= 1.1e+131)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(N[(2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5e+107], t$95$1, If[LessEqual[a, -7.5e-205], t$95$0, If[LessEqual[a, 1.3e-173], N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, 1.1e+131], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.0000000000000002e107 or 1.0999999999999999e131 < a

   1. Initial program 60.1%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 81.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -5.0000000000000002e107 < a < -7.4999999999999996e-205 or 1.30000000000000002e-173 < a < 1.0999999999999999e131

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 95.3

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

   3. Applied rewrites 95.3%

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

   5. Applied rewrites 95.1%

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

   if -7.4999999999999996e-205 < a < 1.30000000000000002e-173

   1. Initial program 71.3%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 80.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 76.1

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

   6. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 76.2

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 94.8

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

   8. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 95.2

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

   10. Applied rewrites 95.2%

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

Alternative 4: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{b - a}{\left(\left(\left(b + a\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b \cdot a\right)}\\ t_1 := \frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b \cdot a}}{b}\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* PI (/ (- b a) (* (* (* (+ b a) 2.0) (- b a)) (* b a)))))
        (t_1 (/ (/ PI a) (* (* a b) 2.0))))
   (if (<= a -2.3e+100)
     t_1
     (if (<= a -8.8e-88)
       t_0
       (if (<= a 6.6e-184)
         (/ (/ (* 0.5 PI) (* b a)) b)
         (if (<= a 4.9e+81) t_0 t_1))))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) * ((b - a) / ((((b + a) * 2.0) * (b - a)) * (b * a)));
	double t_1 = (((double) M_PI) / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -2.3e+100) {
		tmp = t_1;
	} else if (a <= -8.8e-88) {
		tmp = t_0;
	} else if (a <= 6.6e-184) {
		tmp = ((0.5 * ((double) M_PI)) / (b * a)) / b;
	} else if (a <= 4.9e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI * ((b - a) / ((((b + a) * 2.0) * (b - a)) * (b * a)));
	double t_1 = (Math.PI / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -2.3e+100) {
		tmp = t_1;
	} else if (a <= -8.8e-88) {
		tmp = t_0;
	} else if (a <= 6.6e-184) {
		tmp = ((0.5 * Math.PI) / (b * a)) / b;
	} else if (a <= 4.9e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi * ((b - a) / ((((b + a) * 2.0) * (b - a)) * (b * a)))
	t_1 = (math.pi / a) / ((a * b) * 2.0)
	tmp = 0
	if a <= -2.3e+100:
		tmp = t_1
	elif a <= -8.8e-88:
		tmp = t_0
	elif a <= 6.6e-184:
		tmp = ((0.5 * math.pi) / (b * a)) / b
	elif a <= 4.9e+81:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi * Float64(Float64(b - a) / Float64(Float64(Float64(Float64(b + a) * 2.0) * Float64(b - a)) * Float64(b * a))))
	t_1 = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0))
	tmp = 0.0
	if (a <= -2.3e+100)
		tmp = t_1;
	elseif (a <= -8.8e-88)
		tmp = t_0;
	elseif (a <= 6.6e-184)
		tmp = Float64(Float64(Float64(0.5 * pi) / Float64(b * a)) / b);
	elseif (a <= 4.9e+81)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi * ((b - a) / ((((b + a) * 2.0) * (b - a)) * (b * a)));
	t_1 = (pi / a) / ((a * b) * 2.0);
	tmp = 0.0;
	if (a <= -2.3e+100)
		tmp = t_1;
	elseif (a <= -8.8e-88)
		tmp = t_0;
	elseif (a <= 6.6e-184)
		tmp = ((0.5 * pi) / (b * a)) / b;
	elseif (a <= 4.9e+81)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi * N[(N[(b - a), $MachinePrecision] / N[(N[(N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.3e+100], t$95$1, If[LessEqual[a, -8.8e-88], t$95$0, If[LessEqual[a, 6.6e-184], N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, 4.9e+81], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.2999999999999999e100 or 4.90000000000000023e81 < a

   1. Initial program 64.9%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 83.3%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if -2.2999999999999999e100 < a < -8.8000000000000002e-88 or 6.5999999999999995e-184 < a < 4.90000000000000023e81

   1. Initial program 95.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 96.3

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

   3. Applied rewrites 96.3%

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

   5. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. associate-/l/ N/A

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

   9. Applied rewrites 87.2%

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

   if -8.8000000000000002e-88 < a < 6.5999999999999995e-184

   1. Initial program 74.2%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 82.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 75.7

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

   6. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 75.7

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 92.1

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

   8. Applied rewrites 92.1%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.4

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

   10. Applied rewrites 92.4%

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

Alternative 5: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{if}\;a \leq -3 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(a \cdot b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) (* (* a b) 2.0))))
   (if (<= a -3e+16) t_0 (if (<= a 6.6e-9) (/ (* 0.5 PI) (* (* a b) b)) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -3e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (0.5 * ((double) M_PI)) / ((a * b) * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / a) / ((a * b) * 2.0);
	double tmp;
	if (a <= -3e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (0.5 * Math.PI) / ((a * b) * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / ((a * b) * 2.0)
	tmp = 0
	if a <= -3e+16:
		tmp = t_0
	elif a <= 6.6e-9:
		tmp = (0.5 * math.pi) / ((a * b) * b)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0))
	tmp = 0.0
	if (a <= -3e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(a * b) * b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / ((a * b) * 2.0);
	tmp = 0.0;
	if (a <= -3e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = (0.5 * pi) / ((a * b) * b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3e+16], t$95$0, If[LessEqual[a, 6.6e-9], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3e16 or 6.60000000000000037e-9 < a

   1. Initial program 74.9%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 91.8

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

   6. Applied rewrites 91.8%

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

   if -3e16 < a < 6.60000000000000037e-9

   1. Initial program 82.1%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 70.9

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

   6. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 70.9

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 82.3

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

   8. Applied rewrites 82.3%

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

Alternative 6: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a \cdot a}}{b} \cdot 0.5\\ \mathbf{if}\;a \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(a \cdot b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI (* a a)) b) 0.5)))
   (if (<= a -2e+16) t_0 (if (<= a 6.6e-9) (/ (* 0.5 PI) (* (* a b) b)) t_0))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / (a * a)) / b) * 0.5;
	double tmp;
	if (a <= -2e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (0.5 * ((double) M_PI)) / ((a * b) * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / (a * a)) / b) * 0.5;
	double tmp;
	if (a <= -2e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (0.5 * Math.PI) / ((a * b) * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / (a * a)) / b) * 0.5
	tmp = 0
	if a <= -2e+16:
		tmp = t_0
	elif a <= 6.6e-9:
		tmp = (0.5 * math.pi) / ((a * b) * b)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / Float64(a * a)) / b) * 0.5)
	tmp = 0.0
	if (a <= -2e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(a * b) * b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / (a * a)) / b) * 0.5;
	tmp = 0.0;
	if (a <= -2e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = (0.5 * pi) / ((a * b) * b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -2e+16], t$95$0, If[LessEqual[a, 6.6e-9], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2e16 or 6.60000000000000037e-9 < a

   1. Initial program 74.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in a around inf

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

      1. *-lft-identity N/A

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

      2. *-rgt-identity N/A

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

      3. frac-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. difference-of-squares-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 80.1%

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

   if -2e16 < a < 6.60000000000000037e-9

   1. Initial program 82.1%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 70.9

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

   6. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 70.9

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 82.3

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

   8. Applied rewrites 82.3%

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

Alternative 7: 81.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5\\ \mathbf{if}\;a \leq -3 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(a \cdot b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* (* a a) b)) 0.5)))
   (if (<= a -3e+16) t_0 (if (<= a 6.6e-9) (/ (* 0.5 PI) (* (* a b) b)) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / ((a * a) * b)) * 0.5;
	double tmp;
	if (a <= -3e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (0.5 * ((double) M_PI)) / ((a * b) * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / ((a * a) * b)) * 0.5;
	double tmp;
	if (a <= -3e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (0.5 * Math.PI) / ((a * b) * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / ((a * a) * b)) * 0.5
	tmp = 0
	if a <= -3e+16:
		tmp = t_0
	elif a <= 6.6e-9:
		tmp = (0.5 * math.pi) / ((a * b) * b)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(Float64(a * a) * b)) * 0.5)
	tmp = 0.0
	if (a <= -3e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(a * b) * b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / ((a * a) * b)) * 0.5;
	tmp = 0.0;
	if (a <= -3e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = (0.5 * pi) / ((a * b) * b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -3e+16], t$95$0, If[LessEqual[a, 6.6e-9], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3e16 or 6.60000000000000037e-9 < a

   1. Initial program 74.9%

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

   2. 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 80.4

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

   4. Applied rewrites 80.4%

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

   if -3e16 < a < 6.60000000000000037e-9

   1. Initial program 82.1%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 70.9

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

   6. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 70.9

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 82.3

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

   8. Applied rewrites 82.3%

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

Alternative 8: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5\\ \mathbf{if}\;a \leq -3 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* (* a a) b)) 0.5)))
   (if (<= a -3e+16) t_0 (if (<= a 6.6e-9) (* (/ PI (* (* b b) a)) 0.5) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / ((a * a) * b)) * 0.5;
	double tmp;
	if (a <= -3e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / ((a * a) * b)) * 0.5;
	double tmp;
	if (a <= -3e+16) {
		tmp = t_0;
	} else if (a <= 6.6e-9) {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / ((a * a) * b)) * 0.5
	tmp = 0
	if a <= -3e+16:
		tmp = t_0
	elif a <= 6.6e-9:
		tmp = (math.pi / ((b * b) * a)) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(Float64(a * a) * b)) * 0.5)
	tmp = 0.0
	if (a <= -3e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / ((a * a) * b)) * 0.5;
	tmp = 0.0;
	if (a <= -3e+16)
		tmp = t_0;
	elseif (a <= 6.6e-9)
		tmp = (pi / ((b * 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[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -3e+16], t$95$0, If[LessEqual[a, 6.6e-9], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3e16 or 6.60000000000000037e-9 < a

   1. Initial program 74.9%

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

   2. 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 80.4

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

   4. Applied rewrites 80.4%

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

   if -3e16 < a < 6.60000000000000037e-9

   1. Initial program 82.1%

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

   2. 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 70.9

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

   4. Applied rewrites 70.9%

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

Alternative 9: 57.4% accurate, 2.6× speedup

Math

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

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(Float64(a * 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[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

2. 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 57.4

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

4. Applied rewrites 57.4%

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

Reproduce

herbie shell --seed 2025106 
(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))))