NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.7%

Time: 11.2s

Alternatives: 9

Speedup: 2.4×

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: 78.2% 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: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. div-inv N/A

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

   16. lift-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. lft-mult-inverse N/A

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

   10. *-rgt-identity 99.7

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

8. Applied rewrites 99.7%

   \[\leadsto \frac{\frac{\pi \cdot 0.5}{b + a}}{\color{blue}{b \cdot a}} \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi \cdot \frac{0.5}{b + a}}{b \cdot a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. div-inv N/A

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

   16. lift-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. lft-mult-inverse N/A

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

   10. *-rgt-identity 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{\pi \cdot \frac{0.5}{b + a}}{b \cdot a} \]
12. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{b \cdot a} \cdot \frac{\pi}{b + a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. div-inv N/A

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

   16. lift-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l/ N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b + a}} \]
9. Add Preprocessing

Alternative 4: 74.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-5) (/ (* PI 0.5) (* a (* b a))) (/ (* PI 0.5) (* b (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e-5) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (b * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-5) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (Math.PI * 0.5) / (b * (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e-5:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (math.pi * 0.5) / (b * (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e-5)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-5)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (pi * 0.5) / (b * (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e-5], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000016e-5

   1. Initial program 77.5%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -2.00000000000000016e-5 < a

   1. Initial program 82.9%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 76.0

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{b - a} \cdot \frac{\pi \cdot 0.5}{b + a} \]

   7. Applied rewrites 76.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{b - a} \cdot \frac{\pi \cdot 0.5}{b + a} \]

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 65.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{\pi}{b \cdot b}, 0\right)}{a}} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{b \cdot b} + 0}{a} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b \cdot b}} + 0}{a} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{b \cdot b}} + 0}{a} \]

       4. lift-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right)}}{a} \]

       5. --rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right) - 0}}{a} \]

       6. div0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right) - \color{blue}{\frac{0}{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right)}}}{a} \]

       7. --rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right) - \frac{0}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right) - 0}}}{a} \]

       8. --rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right) - \frac{0}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right)}}}{a} \]

       9. div0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right) - \color{blue}{0}}{a} \]

       10. --rgt-identity N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{b \cdot b}, 0\right)}}{a} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b \cdot b} + 0}}{a} \]

       12. +-rgt-identity N/A

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

       13. associate-/l* N/A

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

       14. div-inv N/A

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

       15. associate-*l* N/A

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

   12. Applied rewrites 72.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e-6) (/ (* PI 0.5) (* a (* b a))) (/ (* PI 0.5) (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e-6) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = (((double) M_PI) * 0.5) / (a * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e-6) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (Math.PI * 0.5) / (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e-6:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (math.pi * 0.5) / (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e-6)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e-6)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (pi * 0.5) / (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e-6], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.99999999999999955e-7

   1. Initial program 77.5%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -9.99999999999999955e-7 < a

   1. Initial program 82.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. div-inv N/A

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

   16. lift-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

8. Applied rewrites 98.4%

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

9. Final simplification 98.4%

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

Alternative 7: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. div-inv N/A

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

   16. lift-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l/ N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-*.f64 N/A

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

8. Applied rewrites 98.4%

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

9. Final simplification 98.4%

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

Alternative 8: 63.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 60.9

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

5. Applied rewrites 60.9%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 60.9

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 56.5

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

7. Applied rewrites 56.5%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 60.9

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

9. Applied rewrites 60.9%

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

10. Final simplification 60.9%

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

Alternative 9: 57.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 60.9

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

5. Applied rewrites 60.9%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 60.9

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 56.5

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

7. Applied rewrites 56.5%

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

Reproduce

herbie shell --seed 2024214 
(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))))