NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 11.6s

Alternatives: 8

Speedup: 2.3×

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: 78.5% 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, 2.0× speedup

Math

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

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(pi * Float64(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[(Pi * N[(N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. un-div-inv N/A

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

   2. div-inv N/A

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

   3. difference-of-squares N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. PI-lowering-PI.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. inv-pow N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. +-lowering-+.f64 99.6

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

6. Applied egg-rr 99.6%

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

Alternative 2: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+150)
   (/ (* PI 0.5) (* a (* b a)))
   (* 0.5 (/ PI (* b (* a (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+150) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+150) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+150:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = 0.5 * (math.pi / (b * (a * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+150)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+150)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = 0.5 * (pi / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+150], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.99999999999999981e149

   1. Initial program 45.2%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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. /-lowering-/.f64 N/A

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

      3. +-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 98.9

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{a \cdot \color{blue}{\left(a \cdot b\right)}} \]

   5. Simplified 98.9%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 98.9

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

   7. Applied egg-rr 98.9%

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

   if -9.99999999999999981e149 < a

   1. Initial program 85.1%

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

      1. un-div-inv N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. times-frac N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

      1. *-commutative N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lowering-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative N/A

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

      18. *-lowering-*.f64 N/A

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

      19. +-lowering-+.f64 96.1

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

   8. Applied egg-rr 96.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e-32)
   (/ (* PI 0.5) (* a (* b a)))
   (/ (* PI 0.5) (* b (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-32) {
		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 <= -1.8e-32) {
		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 <= -1.8e-32:
		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 <= -1.8e-32)
		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 <= -1.8e-32)
		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, -1.8e-32], 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 < -1.79999999999999996e-32

   1. Initial program 74.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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. /-lowering-/.f64 N/A

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

      3. +-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 90.3

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{a \cdot \color{blue}{\left(a \cdot b\right)}} \]

   5. Simplified 90.3%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 90.3

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

   7. Applied egg-rr 90.3%

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

   if -1.79999999999999996e-32 < a

   1. Initial program 82.0%

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

      1. un-div-inv N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. times-frac N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.6

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

   9. Simplified 64.6%

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. PI-lowering-PI.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-lowering-*.f64 71.7

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

   11. Applied egg-rr 71.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-32}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.3e-32)
   (* PI (/ 0.5 (* b (* a a))))
   (/ (* PI 0.5) (* b (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.3e-32) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * 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 <= -1.3e-32) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = (Math.PI * 0.5) / (b * (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.3e-32:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = (math.pi * 0.5) / (b * (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.3e-32)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * 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 <= -1.3e-32)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = (pi * 0.5) / (b * (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.3e-32], N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $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 < -1.2999999999999999e-32

   1. Initial program 74.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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. /-lowering-/.f64 N/A

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

      3. +-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 90.3

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{a \cdot \color{blue}{\left(a \cdot b\right)}} \]

   5. Simplified 90.3%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if -1.2999999999999999e-32 < a

   1. Initial program 82.0%

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

      1. un-div-inv N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. times-frac N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.6

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

   9. Simplified 64.6%

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. PI-lowering-PI.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-lowering-*.f64 71.7

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

   11. Applied egg-rr 71.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-32}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-37}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e-37)
   (* PI (/ 0.5 (* b (* a a))))
   (/ (* PI 0.5) (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e-37) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * 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 <= -4.8e-37) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = (Math.PI * 0.5) / (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.8e-37:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = (math.pi * 0.5) / (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.8e-37)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * 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 <= -4.8e-37)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = (pi * 0.5) / (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.8e-37], N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $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 < -4.79999999999999982e-37

   1. Initial program 74.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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. /-lowering-/.f64 N/A

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

      3. +-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 90.3

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{a \cdot \color{blue}{\left(a \cdot b\right)}} \]

   5. Simplified 90.3%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if -4.79999999999999982e-37 < a

   1. Initial program 82.0%

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

      1. un-div-inv N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. times-frac N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. *-lowering-*.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. *-lowering-*.f64 64.6

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

   9. Simplified 64.6%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-37}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e-32)
   (* PI (/ 0.5 (* b (* a a))))
   (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-32) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * 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 <= -1.8e-32) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.8e-32:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.8e-32)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.8e-32)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.8e-32], N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.79999999999999996e-32

   1. Initial program 74.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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. /-lowering-/.f64 N/A

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

      3. +-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 90.3

         \[\leadsto \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{a \cdot \color{blue}{\left(a \cdot b\right)}} \]

   5. Simplified 90.3%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if -1.79999999999999996e-32 < a

   1. Initial program 82.0%

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

      1. un-div-inv N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. times-frac N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.6

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

   9. Simplified 64.6%

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

Alternative 7: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{\left(b \cdot a\right) \cdot \left(b + a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (fma 0.5 PI 0.0) (* (* b a) (+ b a))))

C

double code(double a, double b) {
	return fma(0.5, ((double) M_PI), 0.0) / ((b * a) * (b + a));
}

Julia

function code(a, b)
	return Float64(fma(0.5, pi, 0.0) / Float64(Float64(b * a) * Float64(b + a)))
end

Wolfram

code[a_, b_] := N[(N[(0.5 * Pi + 0.0), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. un-div-inv N/A

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

   2. div-inv N/A

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

   3. difference-of-squares N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. PI-lowering-PI.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

   1. associate-*r/ N/A

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

   2. frac-times N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. inv-pow N/A

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

   6. pow-plus N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. /-lowering-/.f64 N/A

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

   12. +-rgt-identity N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. PI-lowering-PI.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. +-lowering-+.f64 N/A

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

   17. *-lowering-*.f64 99.0

       \[\leadsto \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{\left(b + a\right) \cdot \color{blue}{\left(b \cdot a\right)}} \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{\left(b + a\right) \cdot \left(b \cdot a\right)}} \]

7. Final simplification 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(0.5, \pi, 0\right)}{\left(b \cdot a\right) \cdot \left(b + a\right)} \]
8. Add Preprocessing

Alternative 8: 56.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. un-div-inv N/A

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

   2. div-inv N/A

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

   3. difference-of-squares N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. PI-lowering-PI.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. inv-pow N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. +-lowering-+.f64 99.6

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 58.7

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

9. Simplified 58.7%

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

Reproduce

herbie shell --seed 2024195 
(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))))