NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 3.3s

Alternatives: 7

Speedup: 1.6×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   5. lift--.f64 N/A

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

   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

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

3. Applied rewrites 88.4%

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

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift--.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   5. lift--.f64 N/A

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

   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

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

3. Applied rewrites 88.4%

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

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift-/.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.6× 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 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. 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

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

   5. lift--.f64 N/A

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

   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

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

3. Applied rewrites 88.4%

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

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 4: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{if}\;a \leq -1.85 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI a) (* b a)) 0.5)))
   (if (<= a -1.85e-56)
     t_0
     (if (<= a 5.2e-73) (* (/ PI (* b (* b a))) 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 <= -1.85e-56) {
		tmp = t_0;
	} else if (a <= 5.2e-73) {
		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) / (b * a)) * 0.5;
	double tmp;
	if (a <= -1.85e-56) {
		tmp = t_0;
	} else if (a <= 5.2e-73) {
		tmp = (Math.PI / (b * (b * a))) * 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 <= -1.85e-56:
		tmp = t_0
	elif a <= 5.2e-73:
		tmp = (math.pi / (b * (b * a))) * 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 <= -1.85e-56)
		tmp = t_0;
	elseif (a <= 5.2e-73)
		tmp = Float64(Float64(pi / Float64(b * Float64(b * a))) * 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 <= -1.85e-56)
		tmp = t_0;
	elseif (a <= 5.2e-73)
		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[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[a, -1.85e-56], t$95$0, If[LessEqual[a, 5.2e-73], N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8500000000000001e-56 or 5.2000000000000002e-73 < 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. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 74.0

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

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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. lower-*.f64 82.9

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

   6. Applied rewrites 82.9%

      \[\leadsto \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5 \]
   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 83.3

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

   8. Applied rewrites 83.3%

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

   if -1.8500000000000001e-56 < a < 5.2000000000000002e-73

   1. Initial program 77.9%

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

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

      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

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

      5. lift--.f64 N/A

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

      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 85.5

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

   3. Applied rewrites 85.5%

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

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 73.8

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

   8. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 88.0

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

   10. Applied rewrites 88.0%

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

Alternative 5: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;a \leq -1.85 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* a (* a b))) 0.5)))
   (if (<= a -1.85e-56)
     t_0
     (if (<= a 5.2e-73) (* (/ 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 <= -1.85e-56) {
		tmp = t_0;
	} else if (a <= 5.2e-73) {
		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 <= -1.85e-56) {
		tmp = t_0;
	} else if (a <= 5.2e-73) {
		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 <= -1.85e-56:
		tmp = t_0
	elif a <= 5.2e-73:
		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 <= -1.85e-56)
		tmp = t_0;
	elseif (a <= 5.2e-73)
		tmp = Float64(Float64(pi / Float64(b * Float64(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 <= -1.85e-56)
		tmp = t_0;
	elseif (a <= 5.2e-73)
		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, -1.85e-56], t$95$0, If[LessEqual[a, 5.2e-73], N[(N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8500000000000001e-56 or 5.2000000000000002e-73 < 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. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 74.0

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

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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. lower-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

   if -1.8500000000000001e-56 < a < 5.2000000000000002e-73

   1. Initial program 77.9%

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

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

      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

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

      5. lift--.f64 N/A

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

      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 85.5

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

   3. Applied rewrites 85.5%

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

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 73.8

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

   8. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 88.0

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

   10. Applied rewrites 88.0%

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

Alternative 6: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;a \leq -1.75 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* a (* a b))) 0.5)))
   (if (<= a -1.75e-56)
     t_0
     (if (<= a 4.5e-73) (* (/ 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 <= -1.75e-56) {
		tmp = t_0;
	} else if (a <= 4.5e-73) {
		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 <= -1.75e-56) {
		tmp = t_0;
	} else if (a <= 4.5e-73) {
		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 <= -1.75e-56:
		tmp = t_0
	elif a <= 4.5e-73:
		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 <= -1.75e-56)
		tmp = t_0;
	elseif (a <= 4.5e-73)
		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 <= -1.75e-56)
		tmp = t_0;
	elseif (a <= 4.5e-73)
		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, -1.75e-56], t$95$0, If[LessEqual[a, 4.5e-73], 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 < -1.7499999999999999e-56 or 4.5e-73 < 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. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 74.0

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

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5} \]
   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. lower-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

   if -1.7499999999999999e-56 < a < 4.5e-73

   1. Initial program 77.9%

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

   2. Taylor expanded in a around 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 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 7: 62.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

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 57.4

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

4. Applied rewrites 57.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 62.7

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

6. Applied rewrites 62.7%

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

Reproduce

herbie shell --seed 2025112 
(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))))