NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.7%

Time: 11.0s

Alternatives: 9

Speedup: 1.9×

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: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot b - a \cdot a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   2. difference-of-squares N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{\left(b + a\right) \cdot \left(b - a\right)}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{1 \cdot b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   11. *-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   13. --lowering--.f64 88.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

4. Applied egg-rr 88.6%

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

   1. frac-sub N/A

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

   2. associate-*r/ N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. difference-of-squares N/A

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

   6. *-lft-identity N/A

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

   7. *-rgt-identity N/A

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

   8. associate-/r/ N/A

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

   9. flip-+ N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \color{blue}{\left(a \cdot b\right)}\right) \]

6. Applied egg-rr 99.7%

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

Alternative 2: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{2}}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{2 \cdot b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e-27)
   (/ (/ (/ PI 2.0) a) (* a b))
   (/ (/ PI (* 2.0 b)) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.9e-27) {
		tmp = ((((double) M_PI) / 2.0) / a) / (a * b);
	} else {
		tmp = (((double) M_PI) / (2.0 * b)) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.9e-27) {
		tmp = ((Math.PI / 2.0) / a) / (a * b);
	} else {
		tmp = (Math.PI / (2.0 * b)) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.9e-27:
		tmp = ((math.pi / 2.0) / a) / (a * b)
	else:
		tmp = (math.pi / (2.0 * b)) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.9e-27)
		tmp = Float64(Float64(Float64(pi / 2.0) / a) / Float64(a * b));
	else
		tmp = Float64(Float64(pi / Float64(2.0 * b)) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.9e-27)
		tmp = ((pi / 2.0) / a) / (a * b);
	else
		tmp = (pi / (2.0 * b)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.9e-27], N[(N[(N[(Pi / 2.0), $MachinePrecision] / a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8999999999999998e-27

   1. Initial program 86.4%

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

   3. Taylor expanded in 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

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 78.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 78.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{b}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

      7. /-lowering-/.f64 78.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   7. Applied egg-rr 78.9%

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

      1. associate-/r* N/A

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

      2. frac-times N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{1}{2}\right), \color{blue}{\left(a \cdot b\right)}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{a}\right), \left(\color{blue}{a} \cdot b\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), a\right), \left(\color{blue}{a} \cdot b\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), a\right), \left(a \cdot b\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), a\right), \left(a \cdot b\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), a\right), \left(a \cdot b\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), a\right), \left(a \cdot b\right)\right) \]

      10. *-lowering-*.f64 86.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), a\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 86.1%

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

   if -5.8999999999999998e-27 < a

   1. Initial program 75.5%

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

      1. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot b - a \cdot a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      2. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{\left(b + a\right) \cdot \left(b - a\right)}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{1 \cdot b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      11. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      13. --lowering--.f64 87.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   4. Applied egg-rr 87.3%

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

      1. frac-sub N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. difference-of-squares N/A

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

      6. *-lft-identity N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/r/ N/A

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

      9. flip-+ N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \color{blue}{\left(a \cdot b\right)}\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\left(2 \cdot b\right)}\right), \mathsf{*.f64}\left(a, b\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot 2\right)\right), \mathsf{*.f64}\left(a, b\right)\right) \]

      2. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, 2\right)\right), \mathsf{*.f64}\left(a, b\right)\right) \]

   9. Simplified 70.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{2}}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{2 \cdot b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{2 \cdot b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.1e-20)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (/ (/ PI (* 2.0 b)) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.1e-20) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (((double) M_PI) / (2.0 * b)) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.1e-20) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (Math.PI / (2.0 * b)) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.1e-20:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (math.pi / (2.0 * b)) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.1e-20)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(pi / Float64(2.0 * b)) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.1e-20)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (pi / (2.0 * b)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.1e-20], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.10000000000000019e-20

   1. Initial program 86.3%

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

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 80.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{b}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

      7. /-lowering-/.f64 80.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   7. Applied egg-rr 80.1%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{b}}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\frac{\color{blue}{\frac{\frac{1}{2}}{b}}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\frac{\frac{\color{blue}{\frac{1}{2}}}{b}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{b}\right), \color{blue}{a}\right)\right) \]

      7. /-lowering-/.f64 87.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\right), a\right)\right) \]

   9. Applied egg-rr 87.3%

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

   if -5.10000000000000019e-20 < a

   1. Initial program 75.7%

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

      1. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot b - a \cdot a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      2. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{\left(b + a\right) \cdot \left(b - a\right)}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{1 \cdot b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      11. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

      13. --lowering--.f64 87.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   4. Applied egg-rr 87.4%

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

      1. frac-sub N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. difference-of-squares N/A

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

      6. *-lft-identity N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/r/ N/A

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

      9. flip-+ N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \color{blue}{\left(a \cdot b\right)}\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\left(2 \cdot b\right)}\right), \mathsf{*.f64}\left(a, b\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot 2\right)\right), \mathsf{*.f64}\left(a, b\right)\right) \]

      2. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, 2\right)\right), \mathsf{*.f64}\left(a, b\right)\right) \]

   9. Simplified 70.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{2 \cdot b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{\frac{b}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.6e-20)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (/ (/ PI (* a b)) (/ b 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.6e-20) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (((double) M_PI) / (a * b)) / (b / 0.5);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.6e-20) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (Math.PI / (a * b)) / (b / 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.6e-20:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (math.pi / (a * b)) / (b / 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.6e-20)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) / Float64(b / 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.6e-20)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (pi / (a * b)) / (b / 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.6e-20], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b / 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.6000000000000005e-20

   1. Initial program 86.3%

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

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 80.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{b}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

      7. /-lowering-/.f64 80.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   7. Applied egg-rr 80.1%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{b}}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\frac{\color{blue}{\frac{\frac{1}{2}}{b}}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\frac{\frac{\color{blue}{\frac{1}{2}}}{b}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{b}\right), \color{blue}{a}\right)\right) \]

      7. /-lowering-/.f64 87.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\right), a\right)\right) \]

   9. Applied egg-rr 87.3%

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

   if -5.6000000000000005e-20 < a

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left({b}^{2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{-1}{2}}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{a}\right), \left({b}^{2}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{a}\right), \left({b}^{2}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\frac{\frac{1}{2}}{a}\right)\right)\right), \left({b}^{2}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{1}{2}}{a}\right)\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \left({b}^{2}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

      15. *-lowering-*.f64 55.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   5. Simplified 55.0%

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

   6. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left({b}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{1}{2}\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \frac{1}{2}\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \frac{1}{2}\right), \left({b}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \frac{1}{2}\right), \left({b}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \frac{1}{2}\right), \left(b \cdot \color{blue}{b}\right)\right) \]

      9. *-lowering-*.f64 62.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \frac{1}{2}\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   8. Simplified 62.4%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b}\right), \color{blue}{\left(\frac{b}{\frac{1}{2}}\right)}\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot a}\right), \left(\frac{\color{blue}{b}}{\frac{1}{2}}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot b}\right), \left(\frac{b}{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot b\right)\right), \left(\frac{\color{blue}{b}}{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot b\right)\right), \left(\frac{b}{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{b}{\frac{1}{2}}\right)\right) \]

      10. /-lowering-/.f64 70.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\frac{1}{2}}\right)\right) \]

   10. Applied egg-rr 70.4%

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

Alternative 5: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot b}}{\frac{b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.1e-20)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (/ (/ 0.5 (* a b)) (/ b PI))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.1e-20) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / (a * b)) / (b / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.1e-20) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / (a * b)) / (b / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.1e-20:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (0.5 / (a * b)) / (b / math.pi)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.1e-20)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) / Float64(b / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.1e-20)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (0.5 / (a * b)) / (b / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.1e-20], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.10000000000000019e-20

   1. Initial program 86.3%

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

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 80.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{b}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

      7. /-lowering-/.f64 80.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   7. Applied egg-rr 80.1%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{b}}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\frac{\color{blue}{\frac{\frac{1}{2}}{b}}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\frac{\frac{\color{blue}{\frac{1}{2}}}{b}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{b}\right), \color{blue}{a}\right)\right) \]

      7. /-lowering-/.f64 87.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\right), a\right)\right) \]

   9. Applied egg-rr 87.3%

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

   if -5.10000000000000019e-20 < a

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left({b}^{2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{-1}{2}}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{a}\right), \left({b}^{2}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{a}\right), \left({b}^{2}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\frac{\frac{1}{2}}{a}\right)\right)\right), \left({b}^{2}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{1}{2}}{a}\right)\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \left({b}^{2}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

      15. *-lowering-*.f64 55.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   5. Simplified 55.0%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot \frac{\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}}{b}\right), \color{blue}{\left(\frac{b}{\mathsf{PI}\left(\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \left(\frac{\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}}{b}\right)\right), \left(\frac{\color{blue}{b}}{\mathsf{PI}\left(\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right), b\right)\right), \left(\frac{b}{\mathsf{PI}\left(\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\frac{\frac{1}{2}}{a}\right)\right), b\right)\right), \left(\frac{b}{\mathsf{PI}\left(\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{1}{2}}{a}\right)\right), b\right)\right), \left(\frac{b}{\mathsf{PI}\left(\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right), b\right)\right), \left(\frac{b}{\mathsf{PI}\left(\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right), b\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      11. PI-lowering-PI.f64 63.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right), b\right)\right), \mathsf{/.f64}\left(b, \mathsf{PI.f64}\left(\right)\right)\right) \]

   7. Applied egg-rr 63.1%

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

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}, \mathsf{/.f64}\left(b, \mathsf{PI.f64}\left(\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot b\right)\right), \mathsf{/.f64}\left(\color{blue}{b}, \mathsf{PI.f64}\left(\right)\right)\right) \]

      2. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, b\right)\right), \mathsf{/.f64}\left(b, \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 70.4%

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

Alternative 6: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-20)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (* PI (/ (/ 0.5 (* a b)) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-20) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = ((double) M_PI) * ((0.5 / (a * b)) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-20) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = Math.PI * ((0.5 / (a * b)) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.5e-20:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = math.pi * ((0.5 / (a * b)) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-20)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(pi * Float64(Float64(0.5 / Float64(a * b)) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-20)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = pi * ((0.5 / (a * b)) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.5e-20], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5000000000000001e-20

   1. Initial program 86.3%

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

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 80.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{b}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

      7. /-lowering-/.f64 80.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

   7. Applied egg-rr 80.1%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{b}}{a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\frac{\color{blue}{\frac{\frac{1}{2}}{b}}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\frac{\frac{\color{blue}{\frac{1}{2}}}{b}}{a}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{b}\right), \color{blue}{a}\right)\right) \]

      7. /-lowering-/.f64 87.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\right), a\right)\right) \]

   9. Applied egg-rr 87.3%

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

   if -4.5000000000000001e-20 < a

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left({b}^{2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{-1}{2}}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{a}\right), \left({b}^{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{a}\right), \left({b}^{2}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{b} + \mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{a}\right), \left({b}^{2}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{-1}{2}}{b} + \frac{\frac{1}{2}}{a}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\frac{\frac{1}{2}}{a}\right)\right)\right), \left({b}^{2}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{1}{2}}{a}\right)\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \left({b}^{2}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

      15. *-lowering-*.f64 55.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\frac{1}{2}, a\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   5. Simplified 55.0%

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

   6. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left({b}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{1}{2}\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \frac{1}{2}\right), \left({\color{blue}{b}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \frac{1}{2}\right), \left({b}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \frac{1}{2}\right), \left({b}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \frac{1}{2}\right), \left(b \cdot \color{blue}{b}\right)\right) \]

      9. *-lowering-*.f64 62.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \frac{1}{2}\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   8. Simplified 62.4%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b}\right), \color{blue}{\left(\frac{b}{\frac{1}{2}}\right)}\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b \cdot a}\right), \left(\frac{\color{blue}{b}}{\frac{1}{2}}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot b}\right), \left(\frac{b}{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot b\right)\right), \left(\frac{\color{blue}{b}}{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot b\right)\right), \left(\frac{b}{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \left(\frac{b}{\frac{1}{2}}\right)\right) \]

      10. /-lowering-/.f64 70.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, b\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\frac{1}{2}}\right)\right) \]

   10. Applied egg-rr 70.4%

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

       1. div-inv N/A

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

       2. div-inv N/A

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

       3. clear-num N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot b} \cdot \frac{\frac{1}{2}}{b}\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   12. Applied egg-rr 70.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot b - a \cdot a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   2. difference-of-squares N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{\left(b + a\right) \cdot \left(b - a\right)}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{1 \cdot b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   11. *-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   13. --lowering--.f64 88.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

4. Applied egg-rr 88.6%

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

   1. frac-sub N/A

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

   2. associate-*r/ N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. difference-of-squares N/A

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

   6. *-lft-identity N/A

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

   7. *-rgt-identity N/A

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

   8. associate-/r/ N/A

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

   9. flip-+ N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \color{blue}{\left(a \cdot b\right)}\right) \]

6. Applied egg-rr 99.7%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-/l/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \color{blue}{\left(\left(a \cdot b\right) \cdot \left(a + b\right)\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \left(\left(a \cdot \color{blue}{b}\right) \cdot \left(a + b\right)\right)\right) \]

   7. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(\color{blue}{\left(a \cdot b\right)} \cdot \left(a + b\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(\color{blue}{\left(a \cdot b\right)} \cdot \left(a + b\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\color{blue}{a} \cdot b\right) \cdot \left(a + b\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\left(a \cdot b\right), \color{blue}{\left(a + b\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\color{blue}{a} + b\right)\right)\right) \]

   12. +-lowering-+.f64 99.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{+.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b \cdot b - a \cdot a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   2. difference-of-squares N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{\left(b + a\right) \cdot \left(b - a\right)}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   4. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}}{1 \cdot b - a \cdot 1}\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, a\right)}, \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(b + a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(1 \cdot b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   11. *-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a \cdot 1\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

   13. --lowering--.f64 88.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{a}\right), \mathsf{/.f64}\left(1, b\right)\right)\right) \]

4. Applied egg-rr 88.6%

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

   1. frac-sub N/A

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

   2. associate-*r/ N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. difference-of-squares N/A

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

   6. *-lft-identity N/A

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

   7. *-rgt-identity N/A

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

   8. associate-/r/ N/A

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

   9. flip-+ N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a}\right), \color{blue}{\left(a \cdot b\right)}\right) \]

6. Applied egg-rr 99.7%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-/l/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \color{blue}{\left(\left(a \cdot b\right) \cdot \left(a + b\right)\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \left(\left(a \cdot \color{blue}{b}\right) \cdot \left(a + b\right)\right)\right) \]

   7. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \left(\color{blue}{\left(a \cdot b\right)} \cdot \left(a + b\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \left(\color{blue}{\left(a \cdot b\right)} \cdot \left(a + b\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \left(\left(\color{blue}{a} \cdot b\right) \cdot \left(a + b\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\left(a \cdot b\right), \color{blue}{\left(a + b\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\color{blue}{a} + b\right)\right)\right) \]

   12. +-lowering-+.f64 99.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{+.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

8. Applied egg-rr 99.0%

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

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{\left(a \cdot b\right) \cdot \left(a + b\right)}\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\left(a \cdot b\right) \cdot \left(a + b\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(a \cdot b\right), \left(a + b\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(a + b\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

   10. PI-lowering-PI.f64 99.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

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

Alternative 9: 63.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

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

   2. associate-*r/ N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

   8. *-lowering-*.f64 56.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

5. Simplified 56.2%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a \cdot a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{b}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\frac{1}{2}}{b}\right)\right) \]

   7. /-lowering-/.f64 56.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right) \]

7. Applied egg-rr 56.2%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{b}}{a}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\frac{\color{blue}{\frac{\frac{1}{2}}{b}}}{a}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\frac{\frac{\color{blue}{\frac{1}{2}}}{b}}{a}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{b}\right), \color{blue}{a}\right)\right) \]

   7. /-lowering-/.f64 62.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\right), a\right)\right) \]

9. Applied egg-rr 62.5%

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

Reproduce

herbie shell --seed 2024139 
(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))))