NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.6%

Time: 3.1s

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: 78.0% 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, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{a + b}\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ 1.0 (* a b)) (* PI (/ 0.5 (+ a b)))))

C

double code(double a, double b) {
	return (1.0 / (a * b)) * (((double) M_PI) * (0.5 / (a + b)));
}

Java

public static double code(double a, double b) {
	return (1.0 / (a * b)) * (Math.PI * (0.5 / (a + b)));
}

Python

def code(a, b):
	return (1.0 / (a * b)) * (math.pi * (0.5 / (a + b)))

Julia

function code(a, b)
	return Float64(Float64(1.0 / Float64(a * b)) * Float64(pi * Float64(0.5 / Float64(a + b))))
end

MATLAB

function tmp = code(a, b)
	tmp = (1.0 / (a * b)) * (pi * (0.5 / (a + b)));
end

Wolfram

code[a_, b_] := N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{a \cdot b}} \cdot \frac{0.5 \cdot \pi}{a + b} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.6

      \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \color{blue}{\frac{0.5}{a + b}}\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\left(\pi \cdot \frac{0.5}{a + b}\right)} \]
9. Add Preprocessing

Alternative 2: 76.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+105}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{\left(a + b\right) \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e-68)
   (/ (/ (* PI 0.5) (* a b)) a)
   (if (<= b 1.08e+105)
     (/ (* 0.5 (/ PI a)) (* (+ a b) (- b a)))
     (* (/ 1.0 (* a b)) (* PI (/ 0.5 b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5e-68) {
		tmp = ((((double) M_PI) * 0.5) / (a * b)) / a;
	} else if (b <= 1.08e+105) {
		tmp = (0.5 * (((double) M_PI) / a)) / ((a + b) * (b - a));
	} else {
		tmp = (1.0 / (a * b)) * (((double) M_PI) * (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.5e-68) {
		tmp = ((Math.PI * 0.5) / (a * b)) / a;
	} else if (b <= 1.08e+105) {
		tmp = (0.5 * (Math.PI / a)) / ((a + b) * (b - a));
	} else {
		tmp = (1.0 / (a * b)) * (Math.PI * (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.5e-68:
		tmp = ((math.pi * 0.5) / (a * b)) / a
	elif b <= 1.08e+105:
		tmp = (0.5 * (math.pi / a)) / ((a + b) * (b - a))
	else:
		tmp = (1.0 / (a * b)) * (math.pi * (0.5 / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.5e-68)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / a);
	elseif (b <= 1.08e+105)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(Float64(a + b) * Float64(b - a)));
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(pi * Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.5e-68)
		tmp = ((pi * 0.5) / (a * b)) / a;
	elseif (b <= 1.08e+105)
		tmp = (0.5 * (pi / a)) / ((a + b) * (b - a));
	else
		tmp = (1.0 / (a * b)) * (pi * (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.5e-68], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.08e+105], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.4999999999999997e-68

   1. Initial program 78.0%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.4

         \[\leadsto 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 63.0

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

   6. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if 6.4999999999999997e-68 < b < 1.07999999999999994e105

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.7%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 64.2

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

   6. Applied rewrites 64.2%

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

   if 1.07999999999999994e105 < b

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1}{a \cdot b}} \cdot \frac{0.5 \cdot \pi}{a + b} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.6

         \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \color{blue}{\frac{0.5}{a + b}}\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\left(\pi \cdot \frac{0.5}{a + b}\right)} \]

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 63.6%

       \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{\color{blue}{b}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \frac{0.5 \cdot \pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25e-67)
   (/ (/ (* PI 0.5) (* a b)) a)
   (* (/ 1.0 (* a b)) (/ (* 0.5 PI) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-67) {
		tmp = ((((double) M_PI) * 0.5) / (a * b)) / a;
	} else {
		tmp = (1.0 / (a * b)) * ((0.5 * ((double) M_PI)) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-67) {
		tmp = ((Math.PI * 0.5) / (a * b)) / a;
	} else {
		tmp = (1.0 / (a * b)) * ((0.5 * Math.PI) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.25e-67:
		tmp = ((math.pi * 0.5) / (a * b)) / a
	else:
		tmp = (1.0 / (a * b)) * ((0.5 * math.pi) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25e-67)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / a);
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(Float64(0.5 * pi) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.25e-67)
		tmp = ((pi * 0.5) / (a * b)) / a;
	else
		tmp = (1.0 / (a * b)) * ((0.5 * pi) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25e-67], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * Pi), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25e-67

   1. Initial program 78.0%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.4

         \[\leadsto 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 63.0

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

   6. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if 1.25e-67 < b

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 63.6%

      \[\leadsto \frac{1}{a \cdot b} \cdot \frac{0.5 \cdot \pi}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25e-67)
   (/ (/ (* PI 0.5) (* a b)) a)
   (* (/ 1.0 (* a b)) (* PI (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-67) {
		tmp = ((((double) M_PI) * 0.5) / (a * b)) / a;
	} else {
		tmp = (1.0 / (a * b)) * (((double) M_PI) * (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-67) {
		tmp = ((Math.PI * 0.5) / (a * b)) / a;
	} else {
		tmp = (1.0 / (a * b)) * (Math.PI * (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.25e-67:
		tmp = ((math.pi * 0.5) / (a * b)) / a
	else:
		tmp = (1.0 / (a * b)) * (math.pi * (0.5 / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25e-67)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / a);
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(pi * Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.25e-67)
		tmp = ((pi * 0.5) / (a * b)) / a;
	else
		tmp = (1.0 / (a * b)) * (pi * (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25e-67], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25e-67

   1. Initial program 78.0%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.4

         \[\leadsto 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]

   4. Applied rewrites 57.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 63.0

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

   6. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if 1.25e-67 < b

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1}{a \cdot b}} \cdot \frac{0.5 \cdot \pi}{a + b} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.6

         \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \color{blue}{\frac{0.5}{a + b}}\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\left(\pi \cdot \frac{0.5}{a + b}\right)} \]

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 63.6%

       \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{\color{blue}{b}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 57.4

      \[\leadsto 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]

4. Applied rewrites 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 63.0

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

6. Applied rewrites 63.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 63.1

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

8. Applied rewrites 63.1%

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

Alternative 6: 63.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 57.4

      \[\leadsto 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]

4. Applied rewrites 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 57.4

      \[\leadsto \pi \cdot \frac{0.5}{\color{blue}{{a}^{2} \cdot b}} \]

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 57.4

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

6. Applied rewrites 57.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 63.1

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

8. Applied rewrites 63.1%

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

Alternative 7: 63.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 57.4

      \[\leadsto 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]

4. Applied rewrites 57.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 63.0

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

6. Applied rewrites 63.0%

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

Reproduce

herbie shell --seed 2025155 
(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))))