NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.7% → 99.6%

Time: 4.4s

Alternatives: 7

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: 77.7% 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, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. difference-of-squares N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 87.9

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

4. Applied rewrites 87.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

    1. lift-*.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow-1 N/A

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

    4. lower-/.f64 N/A

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

    5. lift-*.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 2: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.85e+17) (not (<= b 7e-14)))
   (* (/ PI (* b (* b a))) 0.5)
   (/ (/ (* (/ PI b) 0.5) a) a)))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (((double) M_PI) / (b * (b * a))) * 0.5;
	} else {
		tmp = (((((double) M_PI) / b) * 0.5) / a) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (Math.PI / (b * (b * a))) * 0.5;
	} else {
		tmp = (((Math.PI / b) * 0.5) / a) / a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -1.85e+17) or not (b <= 7e-14):
		tmp = (math.pi / (b * (b * a))) * 0.5
	else:
		tmp = (((math.pi / b) * 0.5) / a) / a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -1.85e+17) || !(b <= 7e-14))
		tmp = Float64(Float64(pi / Float64(b * Float64(b * a))) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(pi / b) * 0.5) / a) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -1.85e+17) || ~((b <= 7e-14)))
		tmp = (pi / (b * (b * a))) * 0.5;
	else
		tmp = (((pi / b) * 0.5) / a) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -1.85e+17], N[Not[LessEqual[b, 7e-14]], $MachinePrecision]], N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.85e17 or 7.0000000000000005e-14 < b

   1. Initial program 74.9%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. +-commutative N/A

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

      3. inv-pow N/A

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

      4. inv-pow N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. frac-sub N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 79.1

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

   9. Applied rewrites 79.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 90.5

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

   11. Applied rewrites 90.5%

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

   if -1.85e17 < b < 7.0000000000000005e-14

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\pi}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      11. pow2 N/A

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

      12. lift-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 83.5

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

   10. Applied rewrites 83.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.85e+17) (not (<= b 7e-14)))
   (* (/ PI (* b (* b a))) 0.5)
   (/ (* (/ (/ PI a) b) 0.5) a)))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (((double) M_PI) / (b * (b * a))) * 0.5;
	} else {
		tmp = (((((double) M_PI) / a) / b) * 0.5) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (Math.PI / (b * (b * a))) * 0.5;
	} else {
		tmp = (((Math.PI / a) / b) * 0.5) / a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -1.85e+17) or not (b <= 7e-14):
		tmp = (math.pi / (b * (b * a))) * 0.5
	else:
		tmp = (((math.pi / a) / b) * 0.5) / a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -1.85e+17) || !(b <= 7e-14))
		tmp = Float64(Float64(pi / Float64(b * Float64(b * a))) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) / b) * 0.5) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -1.85e+17) || ~((b <= 7e-14)))
		tmp = (pi / (b * (b * a))) * 0.5;
	else
		tmp = (((pi / a) / b) * 0.5) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -1.85e+17], N[Not[LessEqual[b, 7e-14]], $MachinePrecision]], N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.85e17 or 7.0000000000000005e-14 < b

   1. Initial program 74.9%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. +-commutative N/A

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

      3. inv-pow N/A

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

      4. inv-pow N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. frac-sub N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 79.1

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

   9. Applied rewrites 79.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 90.5

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

   11. Applied rewrites 90.5%

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

   if -1.85e17 < b < 7.0000000000000005e-14

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\pi}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      11. pow2 N/A

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

      12. lift-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 83.5

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

   10. Applied rewrites 83.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.85e+17) (not (<= b 7e-14)))
   (* (/ PI (* b (* b a))) 0.5)
   (* (/ (/ PI a) (* b a)) 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (((double) M_PI) / (b * (b * a))) * 0.5;
	} else {
		tmp = ((((double) M_PI) / a) / (b * a)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (Math.PI / (b * (b * a))) * 0.5;
	} else {
		tmp = ((Math.PI / a) / (b * a)) * 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -1.85e+17) or not (b <= 7e-14):
		tmp = (math.pi / (b * (b * a))) * 0.5
	else:
		tmp = ((math.pi / a) / (b * a)) * 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -1.85e+17) || !(b <= 7e-14))
		tmp = Float64(Float64(pi / Float64(b * Float64(b * a))) * 0.5);
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -1.85e+17) || ~((b <= 7e-14)))
		tmp = (pi / (b * (b * a))) * 0.5;
	else
		tmp = ((pi / a) / (b * a)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -1.85e+17], N[Not[LessEqual[b, 7e-14]], $MachinePrecision]], N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.85e17 or 7.0000000000000005e-14 < b

   1. Initial program 74.9%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. +-commutative N/A

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

      3. inv-pow N/A

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

      4. inv-pow N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. frac-sub N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 79.1

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

   9. Applied rewrites 79.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 90.5

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

   11. Applied rewrites 90.5%

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

   if -1.85e17 < b < 7.0000000000000005e-14

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
   4. 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 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 83.0

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

   7. Applied rewrites 83.0%

      \[\leadsto \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5 \]
   8. 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 83.5

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

   9. Applied rewrites 83.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.85e+17) (not (<= b 7e-14)))
   (* (/ PI (* b (* b a))) 0.5)
   (* (/ PI (* a (* a b))) 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (((double) M_PI) / (b * (b * a))) * 0.5;
	} else {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (Math.PI / (b * (b * a))) * 0.5;
	} else {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -1.85e+17) or not (b <= 7e-14):
		tmp = (math.pi / (b * (b * a))) * 0.5
	else:
		tmp = (math.pi / (a * (a * b))) * 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -1.85e+17) || !(b <= 7e-14))
		tmp = Float64(Float64(pi / Float64(b * Float64(b * a))) * 0.5);
	else
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -1.85e+17) || ~((b <= 7e-14)))
		tmp = (pi / (b * (b * a))) * 0.5;
	else
		tmp = (pi / (a * (a * b))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -1.85e+17], N[Not[LessEqual[b, 7e-14]], $MachinePrecision]], N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.85e17 or 7.0000000000000005e-14 < b

   1. Initial program 74.9%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. +-commutative N/A

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

      3. inv-pow N/A

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

      4. inv-pow N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. frac-sub N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 79.1

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

   9. Applied rewrites 79.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 90.5

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

   11. Applied rewrites 90.5%

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

   if -1.85e17 < b < 7.0000000000000005e-14

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
   4. 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 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 83.0

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

   7. Applied rewrites 83.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.85e+17) (not (<= b 7e-14)))
   (* (/ PI (* (* b b) a)) 0.5)
   (* (/ PI (* a (* a b))) 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	} else {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -1.85e+17) || !(b <= 7e-14)) {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	} else {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -1.85e+17) or not (b <= 7e-14):
		tmp = (math.pi / ((b * b) * a)) * 0.5
	else:
		tmp = (math.pi / (a * (a * b))) * 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -1.85e+17) || !(b <= 7e-14))
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	else
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -1.85e+17) || ~((b <= 7e-14)))
		tmp = (pi / ((b * b) * a)) * 0.5;
	else
		tmp = (pi / (a * (a * b))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -1.85e+17], N[Not[LessEqual[b, 7e-14]], $MachinePrecision]], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.85e17 or 7.0000000000000005e-14 < b

   1. Initial program 74.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -1.85e17 < b < 7.0000000000000005e-14

   1. Initial program 80.4%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
   4. 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 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 83.0

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

   7. Applied rewrites 83.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17} \lor \neg \left(b \leq 7 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% 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 77.7%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
4. 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 57.0

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

5. Applied rewrites 57.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 63.3

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

7. Applied rewrites 63.3%

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

8. Final simplification 63.3%

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

Reproduce

herbie shell --seed 2025086 
(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))))