NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 4.0s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{if}\;a \leq -5 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\left(-a\right) \cdot \pi}{a \cdot b}}{\left(\left(a + b\right) \cdot 2\right) \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI a) (* b a)) 0.5)))
   (if (<= a -5e+121)
     t_0
     (if (<= a 1.25e+100)
       (/ (/ (* (- a) PI) (* a b)) (* (* (+ a b) 2.0) (- a)))
       t_0))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -5e+121) {
		tmp = t_0;
	} else if (a <= 1.25e+100) {
		tmp = ((-a * ((double) M_PI)) / (a * b)) / (((a + b) * 2.0) * -a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -5e+121) {
		tmp = t_0;
	} else if (a <= 1.25e+100) {
		tmp = ((-a * Math.PI) / (a * b)) / (((a + b) * 2.0) * -a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / a) / (b * a)) * 0.5
	tmp = 0
	if a <= -5e+121:
		tmp = t_0
	elif a <= 1.25e+100:
		tmp = ((-a * math.pi) / (a * b)) / (((a + b) * 2.0) * -a)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5)
	tmp = 0.0
	if (a <= -5e+121)
		tmp = t_0;
	elseif (a <= 1.25e+100)
		tmp = Float64(Float64(Float64(Float64(-a) * pi) / Float64(a * b)) / Float64(Float64(Float64(a + b) * 2.0) * Float64(-a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / a) / (b * a)) * 0.5;
	tmp = 0.0;
	if (a <= -5e+121)
		tmp = t_0;
	elseif (a <= 1.25e+100)
		tmp = ((-a * pi) / (a * b)) / (((a + b) * 2.0) * -a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -5e+121], t$95$0, If[LessEqual[a, 1.25e+100], N[(N[(N[((-a) * Pi), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000007e121 or 1.25e100 < a

   1. Initial program 61.2%

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

   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 98.9

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

   6. Applied rewrites 98.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if -5.00000000000000007e121 < a < 1.25e100

   1. Initial program 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-PI.f64 56.3

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

   6. Applied rewrites 56.3%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 89.8

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

   9. Applied rewrites 89.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

   11. Applied rewrites 99.5%

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

Alternative 2: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(-a\right) \cdot \pi}{a \cdot \left(b \cdot \left(\left(\left(a + b\right) \cdot 2\right) \cdot \left(-a\right)\right)\right)}\\ t_1 := \frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (* (- a) PI) (* a (* b (* (* (+ a b) 2.0) (- a))))))
        (t_1 (* (/ (/ PI a) (* b a)) 0.5)))
   (if (<= a -4.6e+150)
     t_1
     (if (<= a -1.95e-134)
       t_0
       (if (<= a 2.9e-156)
         (* (/ PI (* b (* a b))) 0.5)
         (if (<= a 1.1e+91) t_0 t_1))))))

C

double code(double a, double b) {
	double t_0 = (-a * ((double) M_PI)) / (a * (b * (((a + b) * 2.0) * -a)));
	double t_1 = ((((double) M_PI) / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -4.6e+150) {
		tmp = t_1;
	} else if (a <= -1.95e-134) {
		tmp = t_0;
	} else if (a <= 2.9e-156) {
		tmp = (((double) M_PI) / (b * (a * b))) * 0.5;
	} else if (a <= 1.1e+91) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (-a * Math.PI) / (a * (b * (((a + b) * 2.0) * -a)));
	double t_1 = ((Math.PI / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -4.6e+150) {
		tmp = t_1;
	} else if (a <= -1.95e-134) {
		tmp = t_0;
	} else if (a <= 2.9e-156) {
		tmp = (Math.PI / (b * (a * b))) * 0.5;
	} else if (a <= 1.1e+91) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (-a * math.pi) / (a * (b * (((a + b) * 2.0) * -a)))
	t_1 = ((math.pi / a) / (b * a)) * 0.5
	tmp = 0
	if a <= -4.6e+150:
		tmp = t_1
	elif a <= -1.95e-134:
		tmp = t_0
	elif a <= 2.9e-156:
		tmp = (math.pi / (b * (a * b))) * 0.5
	elif a <= 1.1e+91:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(-a) * pi) / Float64(a * Float64(b * Float64(Float64(Float64(a + b) * 2.0) * Float64(-a)))))
	t_1 = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5)
	tmp = 0.0
	if (a <= -4.6e+150)
		tmp = t_1;
	elseif (a <= -1.95e-134)
		tmp = t_0;
	elseif (a <= 2.9e-156)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * b))) * 0.5);
	elseif (a <= 1.1e+91)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (-a * pi) / (a * (b * (((a + b) * 2.0) * -a)));
	t_1 = ((pi / a) / (b * a)) * 0.5;
	tmp = 0.0;
	if (a <= -4.6e+150)
		tmp = t_1;
	elseif (a <= -1.95e-134)
		tmp = t_0;
	elseif (a <= 2.9e-156)
		tmp = (pi / (b * (a * b))) * 0.5;
	elseif (a <= 1.1e+91)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[((-a) * Pi), $MachinePrecision] / N[(a * N[(b * N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -4.6e+150], t$95$1, If[LessEqual[a, -1.95e-134], t$95$0, If[LessEqual[a, 2.9e-156], N[(N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[a, 1.1e+91], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.60000000000000002e150 or 1.1e91 < a

   1. Initial program 59.0%

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

   2. 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 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 98.8

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

   6. Applied rewrites 98.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if -4.60000000000000002e150 < a < -1.95e-134 or 2.90000000000000021e-156 < a < 1.1e91

   1. Initial program 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.3%

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

   4. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-PI.f64 68.5

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

   6. Applied rewrites 68.5%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 92.5

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

   9. Applied rewrites 92.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 92.2

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

       6. lift-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. +-commutative N/A

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

       14. lower-+.f64 92.2

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

   11. Applied rewrites 92.2%

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

   if -1.95e-134 < a < 2.90000000000000021e-156

   1. Initial program 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 83.1

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

   3. Applied rewrites 83.1%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 76.0

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

   6. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 92.1

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

   8. Applied rewrites 92.1%

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

Alternative 3: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-a\right) \cdot \pi}{a \cdot \left(b \cdot \left(\left(\left(a + b\right) \cdot 2\right) \cdot \left(-a\right)\right)\right)}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(b - a\right) \cdot \pi}{\left(\left(a \cdot b\right) \cdot 2\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI a) (* b a)) 0.5)))
   (if (<= a -4.6e+150)
     t_0
     (if (<= a 1.4e-298)
       (/ (* (- a) PI) (* a (* b (* (* (+ a b) 2.0) (- a)))))
       (if (<= a 1.9e+76)
         (/ (* (- b a) PI) (* (* (* a b) 2.0) (* (- b a) (+ a b))))
         t_0)))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -4.6e+150) {
		tmp = t_0;
	} else if (a <= 1.4e-298) {
		tmp = (-a * ((double) M_PI)) / (a * (b * (((a + b) * 2.0) * -a)));
	} else if (a <= 1.9e+76) {
		tmp = ((b - a) * ((double) M_PI)) / (((a * b) * 2.0) * ((b - a) * (a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -4.6e+150) {
		tmp = t_0;
	} else if (a <= 1.4e-298) {
		tmp = (-a * Math.PI) / (a * (b * (((a + b) * 2.0) * -a)));
	} else if (a <= 1.9e+76) {
		tmp = ((b - a) * Math.PI) / (((a * b) * 2.0) * ((b - a) * (a + b)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / a) / (b * a)) * 0.5
	tmp = 0
	if a <= -4.6e+150:
		tmp = t_0
	elif a <= 1.4e-298:
		tmp = (-a * math.pi) / (a * (b * (((a + b) * 2.0) * -a)))
	elif a <= 1.9e+76:
		tmp = ((b - a) * math.pi) / (((a * b) * 2.0) * ((b - a) * (a + b)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5)
	tmp = 0.0
	if (a <= -4.6e+150)
		tmp = t_0;
	elseif (a <= 1.4e-298)
		tmp = Float64(Float64(Float64(-a) * pi) / Float64(a * Float64(b * Float64(Float64(Float64(a + b) * 2.0) * Float64(-a)))));
	elseif (a <= 1.9e+76)
		tmp = Float64(Float64(Float64(b - a) * pi) / Float64(Float64(Float64(a * b) * 2.0) * Float64(Float64(b - a) * Float64(a + b))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / a) / (b * a)) * 0.5;
	tmp = 0.0;
	if (a <= -4.6e+150)
		tmp = t_0;
	elseif (a <= 1.4e-298)
		tmp = (-a * pi) / (a * (b * (((a + b) * 2.0) * -a)));
	elseif (a <= 1.9e+76)
		tmp = ((b - a) * pi) / (((a * b) * 2.0) * ((b - a) * (a + b)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -4.6e+150], t$95$0, If[LessEqual[a, 1.4e-298], N[(N[((-a) * Pi), $MachinePrecision] / N[(a * N[(b * N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.9e+76], N[(N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.60000000000000002e150 or 1.90000000000000012e76 < a

   1. Initial program 60.7%

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

   2. 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 81.2

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

   4. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -4.60000000000000002e150 < a < 1.39999999999999996e-298

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-PI.f64 58.6

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

   6. Applied rewrites 58.6%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 88.8

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

   9. Applied rewrites 88.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 88.6

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

       6. lift-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. +-commutative N/A

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

       14. lower-+.f64 88.6

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

   11. Applied rewrites 88.6%

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

   if 1.39999999999999996e-298 < a < 1.90000000000000012e76

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 90.5

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

   3. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

   5. Applied rewrites 84.8%

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

Alternative 4: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-a\right) \cdot \pi}{a \cdot \left(b \cdot \left(\left(\left(a + b\right) \cdot 2\right) \cdot \left(-a\right)\right)\right)}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\left(b - a\right) \cdot \frac{\pi}{\left(\left(a \cdot b\right) \cdot 2\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI a) (* b a)) 0.5)))
   (if (<= a -4.6e+150)
     t_0
     (if (<= a 1.4e-298)
       (/ (* (- a) PI) (* a (* b (* (* (+ a b) 2.0) (- a)))))
       (if (<= a 2e+80)
         (* (- b a) (/ PI (* (* (* a b) 2.0) (* (- b a) (+ a b)))))
         t_0)))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -4.6e+150) {
		tmp = t_0;
	} else if (a <= 1.4e-298) {
		tmp = (-a * ((double) M_PI)) / (a * (b * (((a + b) * 2.0) * -a)));
	} else if (a <= 2e+80) {
		tmp = (b - a) * (((double) M_PI) / (((a * b) * 2.0) * ((b - a) * (a + b))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -4.6e+150) {
		tmp = t_0;
	} else if (a <= 1.4e-298) {
		tmp = (-a * Math.PI) / (a * (b * (((a + b) * 2.0) * -a)));
	} else if (a <= 2e+80) {
		tmp = (b - a) * (Math.PI / (((a * b) * 2.0) * ((b - a) * (a + b))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / a) / (b * a)) * 0.5
	tmp = 0
	if a <= -4.6e+150:
		tmp = t_0
	elif a <= 1.4e-298:
		tmp = (-a * math.pi) / (a * (b * (((a + b) * 2.0) * -a)))
	elif a <= 2e+80:
		tmp = (b - a) * (math.pi / (((a * b) * 2.0) * ((b - a) * (a + b))))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5)
	tmp = 0.0
	if (a <= -4.6e+150)
		tmp = t_0;
	elseif (a <= 1.4e-298)
		tmp = Float64(Float64(Float64(-a) * pi) / Float64(a * Float64(b * Float64(Float64(Float64(a + b) * 2.0) * Float64(-a)))));
	elseif (a <= 2e+80)
		tmp = Float64(Float64(b - a) * Float64(pi / Float64(Float64(Float64(a * b) * 2.0) * Float64(Float64(b - a) * Float64(a + b)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / a) / (b * a)) * 0.5;
	tmp = 0.0;
	if (a <= -4.6e+150)
		tmp = t_0;
	elseif (a <= 1.4e-298)
		tmp = (-a * pi) / (a * (b * (((a + b) * 2.0) * -a)));
	elseif (a <= 2e+80)
		tmp = (b - a) * (pi / (((a * b) * 2.0) * ((b - a) * (a + b))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -4.6e+150], t$95$0, If[LessEqual[a, 1.4e-298], N[(N[((-a) * Pi), $MachinePrecision] / N[(a * N[(b * N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2e+80], N[(N[(b - a), $MachinePrecision] * N[(Pi / N[(N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.60000000000000002e150 or 2e80 < a

   1. Initial program 60.1%

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

   2. 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 81.0

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

   4. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -4.60000000000000002e150 < a < 1.39999999999999996e-298

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-PI.f64 58.6

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

   6. Applied rewrites 58.6%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 88.8

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

   9. Applied rewrites 88.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 88.6

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

       6. lift-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. +-commutative N/A

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

       14. lower-+.f64 88.6

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

   11. Applied rewrites 88.6%

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

   if 1.39999999999999996e-298 < a < 2e80

   1. Initial program 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 90.6

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

   3. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

   5. Applied rewrites 84.7%

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

Alternative 5: 88.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-\pi}{\left(b \cdot \left(\left(b + a\right) \cdot 2\right)\right) \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7e+17)
   (* (/ (/ PI a) (* b a)) 0.5)
   (if (<= a 5.8e-68)
     (* (/ PI (* b (* a b))) 0.5)
     (/ (- PI) (* (* b (* (+ b a) 2.0)) (- b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7e+17) {
		tmp = ((((double) M_PI) / a) / (b * a)) * 0.5;
	} else if (a <= 5.8e-68) {
		tmp = (((double) M_PI) / (b * (a * b))) * 0.5;
	} else {
		tmp = -((double) M_PI) / ((b * ((b + a) * 2.0)) * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7e+17) {
		tmp = ((Math.PI / a) / (b * a)) * 0.5;
	} else if (a <= 5.8e-68) {
		tmp = (Math.PI / (b * (a * b))) * 0.5;
	} else {
		tmp = -Math.PI / ((b * ((b + a) * 2.0)) * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7e+17:
		tmp = ((math.pi / a) / (b * a)) * 0.5
	elif a <= 5.8e-68:
		tmp = (math.pi / (b * (a * b))) * 0.5
	else:
		tmp = -math.pi / ((b * ((b + a) * 2.0)) * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7e+17)
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5);
	elseif (a <= 5.8e-68)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * b))) * 0.5);
	else
		tmp = Float64(Float64(-pi) / Float64(Float64(b * Float64(Float64(b + a) * 2.0)) * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7e+17)
		tmp = ((pi / a) / (b * a)) * 0.5;
	elseif (a <= 5.8e-68)
		tmp = (pi / (b * (a * b))) * 0.5;
	else
		tmp = -pi / ((b * ((b + a) * 2.0)) * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7e+17], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[a, 5.8e-68], N[(N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[((-Pi) / N[(N[(b * N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7e17

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 82.2

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

   4. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

   if -7e17 < a < 5.8000000000000001e-68

   1. Initial program 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 87.4

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

   3. Applied rewrites 87.4%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 84.5

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

   8. Applied rewrites 84.5%

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

   if 5.8000000000000001e-68 < a

   1. Initial program 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 89.5

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

   3. Applied rewrites 89.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-PI.f64 89.5

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift--.f64 89.5

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

   5. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. frac-sub N/A

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

      13. *-rgt-identity N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lift-PI.f64 90.1

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

   10. Applied rewrites 90.1%

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

Alternative 6: 86.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{if}\;a \leq -7 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI a) (* b a)) 0.5)))
   (if (<= a -7e+17)
     t_0
     (if (<= a 1.3e-67) (* (/ PI (* b (* a b))) 0.5) t_0))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -7e+17) {
		tmp = t_0;
	} else if (a <= 1.3e-67) {
		tmp = (((double) M_PI) / (b * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / a) / (b * a)) * 0.5;
	double tmp;
	if (a <= -7e+17) {
		tmp = t_0;
	} else if (a <= 1.3e-67) {
		tmp = (Math.PI / (b * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / a) / (b * a)) * 0.5
	tmp = 0
	if a <= -7e+17:
		tmp = t_0
	elif a <= 1.3e-67:
		tmp = (math.pi / (b * (a * b))) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5)
	tmp = 0.0
	if (a <= -7e+17)
		tmp = t_0;
	elseif (a <= 1.3e-67)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * b))) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / a) / (b * a)) * 0.5;
	tmp = 0.0;
	if (a <= -7e+17)
		tmp = t_0;
	elseif (a <= 1.3e-67)
		tmp = (pi / (b * (a * b))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -7e+17], t$95$0, If[LessEqual[a, 1.3e-67], N[(N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -7e17 or 1.2999999999999999e-67 < a

   1. Initial program 76.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 77.6

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

   4. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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 87.6

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

   6. Applied rewrites 87.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if -7e17 < a < 1.2999999999999999e-67

   1. Initial program 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 87.4

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

   3. Applied rewrites 87.4%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 84.6

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

   8. Applied rewrites 84.6%

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

Alternative 7: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;a \leq -7 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* a (* a b))) 0.5)))
   (if (<= a -7e+17)
     t_0
     (if (<= a 1.3e-67) (* (/ PI (* b (* a b))) 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 <= -7e+17) {
		tmp = t_0;
	} else if (a <= 1.3e-67) {
		tmp = (((double) M_PI) / (b * (a * b))) * 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 <= -7e+17) {
		tmp = t_0;
	} else if (a <= 1.3e-67) {
		tmp = (Math.PI / (b * (a * b))) * 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 <= -7e+17:
		tmp = t_0
	elif a <= 1.3e-67:
		tmp = (math.pi / (b * (a * b))) * 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 <= -7e+17)
		tmp = t_0;
	elseif (a <= 1.3e-67)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * b))) * 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 <= -7e+17)
		tmp = t_0;
	elseif (a <= 1.3e-67)
		tmp = (pi / (b * (a * b))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -7e+17], t$95$0, If[LessEqual[a, 1.3e-67], N[(N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -7e17 or 1.2999999999999999e-67 < a

   1. Initial program 76.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 77.6

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

   4. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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 87.6

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

   6. Applied rewrites 87.6%

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

   if -7e17 < a < 1.2999999999999999e-67

   1. Initial program 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 87.4

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

   3. Applied rewrites 87.4%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 84.6

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

   8. Applied rewrites 84.6%

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

Alternative 8: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;a \leq -7 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* a (* a b))) 0.5)))
   (if (<= a -7e+17)
     t_0
     (if (<= a 1.3e-67) (* (/ 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 <= -7e+17) {
		tmp = t_0;
	} else if (a <= 1.3e-67) {
		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 <= -7e+17) {
		tmp = t_0;
	} else if (a <= 1.3e-67) {
		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 <= -7e+17:
		tmp = t_0
	elif a <= 1.3e-67:
		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 <= -7e+17)
		tmp = t_0;
	elseif (a <= 1.3e-67)
		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 <= -7e+17)
		tmp = t_0;
	elseif (a <= 1.3e-67)
		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, -7e+17], t$95$0, If[LessEqual[a, 1.3e-67], 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 < -7e17 or 1.2999999999999999e-67 < a

   1. Initial program 76.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 77.6

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

   4. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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 87.6

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

   6. Applied rewrites 87.6%

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

   if -7e17 < a < 1.2999999999999999e-67

   1. Initial program 81.4%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 72.8

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

   4. Applied rewrites 72.8%

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

Alternative 9: 62.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

2. 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 57.5

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

4. Applied rewrites 57.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 62.9

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

6. Applied rewrites 62.9%

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

Reproduce

herbie shell --seed 2025105 
(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))))