NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.1%

Time: 4.1s

Alternatives: 8

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   5. lift--.f64 N/A

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

   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 88.4

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

3. Applied rewrites 88.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lift--.f64 88.5

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

7. Applied rewrites 88.5%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. frac-times N/A

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

   12. lower-/.f64 N/A

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

9. Applied rewrites 99.1%

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

Alternative 2: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a}{a \cdot b}\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;\frac{\pi \cdot t\_0}{\left(\left(a + b\right) \cdot 2\right) \cdot \left(b - a\right)}\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b \cdot a\right) \cdot b}\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+130}:\\ \;\;\;\;\frac{\pi}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (- b a) (* a b))))
   (if (<= a -6.6e+154)
     (* (/ PI (* a (* a b))) 0.5)
     (if (<= a -1.45e-198)
       (/ (* PI t_0) (* (* (+ a b) 2.0) (- b a)))
       (if (<= a 9.2e-174)
         (/ (* 0.5 PI) (* (* b a) b))
         (if (<= a 7e+130)
           (* (/ PI (* (* 2.0 (+ b a)) (- b a))) t_0)
           (/ (/ PI a) (* (* a b) 2.0))))))))

C

double code(double a, double b) {
	double t_0 = (b - a) / (a * b);
	double tmp;
	if (a <= -6.6e+154) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else if (a <= -1.45e-198) {
		tmp = (((double) M_PI) * t_0) / (((a + b) * 2.0) * (b - a));
	} else if (a <= 9.2e-174) {
		tmp = (0.5 * ((double) M_PI)) / ((b * a) * b);
	} else if (a <= 7e+130) {
		tmp = (((double) M_PI) / ((2.0 * (b + a)) * (b - a))) * t_0;
	} else {
		tmp = (((double) M_PI) / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (b - a) / (a * b);
	double tmp;
	if (a <= -6.6e+154) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else if (a <= -1.45e-198) {
		tmp = (Math.PI * t_0) / (((a + b) * 2.0) * (b - a));
	} else if (a <= 9.2e-174) {
		tmp = (0.5 * Math.PI) / ((b * a) * b);
	} else if (a <= 7e+130) {
		tmp = (Math.PI / ((2.0 * (b + a)) * (b - a))) * t_0;
	} else {
		tmp = (Math.PI / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b - a) / (a * b)
	tmp = 0
	if a <= -6.6e+154:
		tmp = (math.pi / (a * (a * b))) * 0.5
	elif a <= -1.45e-198:
		tmp = (math.pi * t_0) / (((a + b) * 2.0) * (b - a))
	elif a <= 9.2e-174:
		tmp = (0.5 * math.pi) / ((b * a) * b)
	elif a <= 7e+130:
		tmp = (math.pi / ((2.0 * (b + a)) * (b - a))) * t_0
	else:
		tmp = (math.pi / a) / ((a * b) * 2.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b - a) / Float64(a * b))
	tmp = 0.0
	if (a <= -6.6e+154)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	elseif (a <= -1.45e-198)
		tmp = Float64(Float64(pi * t_0) / Float64(Float64(Float64(a + b) * 2.0) * Float64(b - a)));
	elseif (a <= 9.2e-174)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(b * a) * b));
	elseif (a <= 7e+130)
		tmp = Float64(Float64(pi / Float64(Float64(2.0 * Float64(b + a)) * Float64(b - a))) * t_0);
	else
		tmp = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b - a) / (a * b);
	tmp = 0.0;
	if (a <= -6.6e+154)
		tmp = (pi / (a * (a * b))) * 0.5;
	elseif (a <= -1.45e-198)
		tmp = (pi * t_0) / (((a + b) * 2.0) * (b - a));
	elseif (a <= 9.2e-174)
		tmp = (0.5 * pi) / ((b * a) * b);
	elseif (a <= 7e+130)
		tmp = (pi / ((2.0 * (b + a)) * (b - a))) * t_0;
	else
		tmp = (pi / a) / ((a * b) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.6e+154], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[a, -1.45e-198], N[(N[(Pi * t$95$0), $MachinePrecision] / N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.2e-174], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7e+130], N[(N[(Pi / N[(N[(2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -6.6e154

   1. Initial program 50.2%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -6.6e154 < a < -1.45e-198

   1. Initial program 92.9%

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

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

      5. lift--.f64 N/A

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

      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 95.6

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

   3. Applied rewrites 95.6%

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

   5. Applied rewrites 95.2%

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

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

      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

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

      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-*l/ N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 95.2%

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

   if -1.45e-198 < a < 9.1999999999999995e-174

   1. Initial program 71.2%

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

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

      8. lift--.f64 N/A

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

      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%

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

   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 76.0

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

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}} \]
   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 76.1

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      13. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

   if 9.1999999999999995e-174 < a < 7.0000000000000002e130

   1. Initial program 94.7%

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

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

      5. lift--.f64 N/A

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

      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%

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

   if 7.0000000000000002e130 < a

   1. Initial program 57.0%

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

      1. lift-*.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 79.8%

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

   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%

      \[\leadsto \frac{\color{blue}{\frac{\pi}{a}}}{\left(a \cdot b\right) \cdot 2} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \cdot \frac{b - a}{a \cdot b}\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b \cdot a\right) \cdot b}\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* (* 2.0 (+ b a)) (- b a))) (/ (- b a) (* a b)))))
   (if (<= a -6.6e+154)
     (* (/ PI (* a (* a b))) 0.5)
     (if (<= a -1.45e-198)
       t_0
       (if (<= a 9.2e-174)
         (/ (* 0.5 PI) (* (* b a) b))
         (if (<= a 7e+130) t_0 (/ (/ PI a) (* (* a b) 2.0))))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	double tmp;
	if (a <= -6.6e+154) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else if (a <= -1.45e-198) {
		tmp = t_0;
	} else if (a <= 9.2e-174) {
		tmp = (0.5 * ((double) M_PI)) / ((b * a) * b);
	} else if (a <= 7e+130) {
		tmp = t_0;
	} else {
		tmp = (((double) M_PI) / a) / ((a * b) * 2.0);
	}
	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 tmp;
	if (a <= -6.6e+154) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else if (a <= -1.45e-198) {
		tmp = t_0;
	} else if (a <= 9.2e-174) {
		tmp = (0.5 * Math.PI) / ((b * a) * b);
	} else if (a <= 7e+130) {
		tmp = t_0;
	} else {
		tmp = (Math.PI / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b))
	tmp = 0
	if a <= -6.6e+154:
		tmp = (math.pi / (a * (a * b))) * 0.5
	elif a <= -1.45e-198:
		tmp = t_0
	elif a <= 9.2e-174:
		tmp = (0.5 * math.pi) / ((b * a) * b)
	elif a <= 7e+130:
		tmp = t_0
	else:
		tmp = (math.pi / a) / ((a * b) * 2.0)
	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)))
	tmp = 0.0
	if (a <= -6.6e+154)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	elseif (a <= -1.45e-198)
		tmp = t_0;
	elseif (a <= 9.2e-174)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(b * a) * b));
	elseif (a <= 7e+130)
		tmp = t_0;
	else
		tmp = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / ((2.0 * (b + a)) * (b - a))) * ((b - a) / (a * b));
	tmp = 0.0;
	if (a <= -6.6e+154)
		tmp = (pi / (a * (a * b))) * 0.5;
	elseif (a <= -1.45e-198)
		tmp = t_0;
	elseif (a <= 9.2e-174)
		tmp = (0.5 * pi) / ((b * a) * b);
	elseif (a <= 7e+130)
		tmp = t_0;
	else
		tmp = (pi / a) / ((a * b) * 2.0);
	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]}, If[LessEqual[a, -6.6e+154], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[a, -1.45e-198], t$95$0, If[LessEqual[a, 9.2e-174], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7e+130], t$95$0, N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.6e154

   1. Initial program 50.2%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -6.6e154 < a < -1.45e-198 or 9.1999999999999995e-174 < a < 7.0000000000000002e130

   1. Initial program 93.7%

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

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

      5. lift--.f64 N/A

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

      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 95.7

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

   3. Applied rewrites 95.7%

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

   5. Applied rewrites 95.4%

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

   if -1.45e-198 < a < 9.1999999999999995e-174

   1. Initial program 71.2%

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

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

      8. lift--.f64 N/A

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

      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%

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

   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 76.0

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

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}} \]
   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 76.1

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      13. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

   if 7.0000000000000002e130 < a

   1. Initial program 57.0%

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

      1. lift-*.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 79.8%

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

   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%

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

Alternative 4: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi \cdot \left(b - a\right)}{\left(\left(\left(a + b\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(a \cdot b\right)}\\ \mathbf{if}\;a \leq -6.3 \cdot 10^{+94}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{-185}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b \cdot a\right) \cdot b}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (* PI (- b a)) (* (* (* (+ a b) 2.0) (- b a)) (* a b)))))
   (if (<= a -6.3e+94)
     (* (/ PI (* a (* a b))) 0.5)
     (if (<= a -5.8e-88)
       t_0
       (if (<= a 1.32e-185)
         (/ (* 0.5 PI) (* (* b a) b))
         (if (<= a 4.8e+81) t_0 (/ (/ PI a) (* (* a b) 2.0))))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) * (b - a)) / ((((a + b) * 2.0) * (b - a)) * (a * b));
	double tmp;
	if (a <= -6.3e+94) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else if (a <= -5.8e-88) {
		tmp = t_0;
	} else if (a <= 1.32e-185) {
		tmp = (0.5 * ((double) M_PI)) / ((b * a) * b);
	} else if (a <= 4.8e+81) {
		tmp = t_0;
	} else {
		tmp = (((double) M_PI) / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI * (b - a)) / ((((a + b) * 2.0) * (b - a)) * (a * b));
	double tmp;
	if (a <= -6.3e+94) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else if (a <= -5.8e-88) {
		tmp = t_0;
	} else if (a <= 1.32e-185) {
		tmp = (0.5 * Math.PI) / ((b * a) * b);
	} else if (a <= 4.8e+81) {
		tmp = t_0;
	} else {
		tmp = (Math.PI / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi * (b - a)) / ((((a + b) * 2.0) * (b - a)) * (a * b))
	tmp = 0
	if a <= -6.3e+94:
		tmp = (math.pi / (a * (a * b))) * 0.5
	elif a <= -5.8e-88:
		tmp = t_0
	elif a <= 1.32e-185:
		tmp = (0.5 * math.pi) / ((b * a) * b)
	elif a <= 4.8e+81:
		tmp = t_0
	else:
		tmp = (math.pi / a) / ((a * b) * 2.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi * Float64(b - a)) / Float64(Float64(Float64(Float64(a + b) * 2.0) * Float64(b - a)) * Float64(a * b)))
	tmp = 0.0
	if (a <= -6.3e+94)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	elseif (a <= -5.8e-88)
		tmp = t_0;
	elseif (a <= 1.32e-185)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(b * a) * b));
	elseif (a <= 4.8e+81)
		tmp = t_0;
	else
		tmp = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi * (b - a)) / ((((a + b) * 2.0) * (b - a)) * (a * b));
	tmp = 0.0;
	if (a <= -6.3e+94)
		tmp = (pi / (a * (a * b))) * 0.5;
	elseif (a <= -5.8e-88)
		tmp = t_0;
	elseif (a <= 1.32e-185)
		tmp = (0.5 * pi) / ((b * a) * b);
	elseif (a <= 4.8e+81)
		tmp = t_0;
	else
		tmp = (pi / a) / ((a * b) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.3e+94], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[a, -5.8e-88], t$95$0, If[LessEqual[a, 1.32e-185], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.8e+81], t$95$0, N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.3000000000000001e94

   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. 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 82.9

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

   4. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 99.0

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

   6. Applied rewrites 99.0%

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

   if -6.3000000000000001e94 < a < -5.8000000000000003e-88 or 1.32e-185 < a < 4.79999999999999979e81

   1. Initial program 95.0%

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

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

      5. lift--.f64 N/A

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

      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 96.2

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

   3. Applied rewrites 96.2%

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

   5. Applied rewrites 96.0%

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

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

      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

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

      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. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 87.2%

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

   if -5.8000000000000003e-88 < a < 1.32e-185

   1. Initial program 74.1%

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

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

      8. lift--.f64 N/A

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

      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%

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

   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 75.6

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

   6. Applied rewrites 75.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}} \]
   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 75.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      13. lower-*.f64 92.0

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

   8. Applied rewrites 92.0%

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

   if 4.79999999999999979e81 < a

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      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

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

      8. lift--.f64 N/A

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

      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 83.9%

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

   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 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 5: 87.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+16)
   (* (/ PI (* a (* a b))) 0.5)
   (if (<= a 1.85e-10)
     (/ (* 0.5 PI) (* (* b a) b))
     (/ (/ PI a) (* (* a b) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+16) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else if (a <= 1.85e-10) {
		tmp = (0.5 * ((double) M_PI)) / ((b * a) * b);
	} else {
		tmp = (((double) M_PI) / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e+16) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else if (a <= 1.85e-10) {
		tmp = (0.5 * Math.PI) / ((b * a) * b);
	} else {
		tmp = (Math.PI / a) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e+16:
		tmp = (math.pi / (a * (a * b))) * 0.5
	elif a <= 1.85e-10:
		tmp = (0.5 * math.pi) / ((b * a) * b)
	else:
		tmp = (math.pi / a) / ((a * b) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+16)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	elseif (a <= 1.85e-10)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(b * a) * b));
	else
		tmp = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+16)
		tmp = (pi / (a * (a * b))) * 0.5;
	elseif (a <= 1.85e-10)
		tmp = (0.5 * pi) / ((b * a) * b);
	else
		tmp = (pi / a) / ((a * b) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+16], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[a, 1.85e-10], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2e16

   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. 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 81.9

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

   4. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 93.7

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

   6. Applied rewrites 93.7%

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

   if -2e16 < a < 1.85000000000000007e-10

   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

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

      8. lift--.f64 N/A

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

      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 87.7%

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

   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 71.0

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   8. Applied rewrites 82.3%

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

   if 1.85000000000000007e-10 < a

   1. Initial program 75.7%

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

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

      8. lift--.f64 N/A

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

      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 88.4%

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

   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 89.8

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

   6. Applied rewrites 89.8%

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

Alternative 6: 86.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;a \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* a (* a b))) 0.5)))
   (if (<= a -2e+16)
     t_0
     (if (<= a 1.85e-10) (/ (* 0.5 PI) (* (* b a) 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 <= 1.85e-10) {
		tmp = (0.5 * ((double) M_PI)) / ((b * 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))) * 0.5;
	double tmp;
	if (a <= -2e+16) {
		tmp = t_0;
	} else if (a <= 1.85e-10) {
		tmp = (0.5 * Math.PI) / ((b * a) * 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 <= 1.85e-10:
		tmp = (0.5 * math.pi) / ((b * a) * b)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5)
	tmp = 0.0
	if (a <= -2e+16)
		tmp = t_0;
	elseif (a <= 1.85e-10)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(b * a) * 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 <= 1.85e-10)
		tmp = (0.5 * pi) / ((b * a) * b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -2e+16], t$95$0, If[LessEqual[a, 1.85e-10], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2e16 or 1.85000000000000007e-10 < 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 80.3

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

   4. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 91.6

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

   6. Applied rewrites 91.6%

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

   if -2e16 < a < 1.85000000000000007e-10

   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

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

      8. lift--.f64 N/A

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

      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 87.7%

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

   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 71.0

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   8. Applied rewrites 82.3%

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

Alternative 7: 81.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;a \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* a (* a b))) 0.5)))
   (if (<= a -2e+16)
     t_0
     (if (<= a 1.85e-10) (* (/ 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 <= -2e+16) {
		tmp = t_0;
	} else if (a <= 1.85e-10) {
		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 <= -2e+16) {
		tmp = t_0;
	} else if (a <= 1.85e-10) {
		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 <= -2e+16:
		tmp = t_0
	elif a <= 1.85e-10:
		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(a * Float64(a * b))) * 0.5)
	tmp = 0.0
	if (a <= -2e+16)
		tmp = t_0;
	elseif (a <= 1.85e-10)
		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 <= -2e+16)
		tmp = t_0;
	elseif (a <= 1.85e-10)
		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[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -2e+16], t$95$0, If[LessEqual[a, 1.85e-10], 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 < -2e16 or 1.85000000000000007e-10 < 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 80.3

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

   4. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 91.6

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

   6. Applied rewrites 91.6%

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

   if -2e16 < a < 1.85000000000000007e-10

   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 8: 63.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 63.0

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

6. Applied rewrites 63.0%

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