NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.7%

Time: 12.1s

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: 78.7% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-out-- N/A

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

   5. div-inv N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. difference-of-squares N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

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

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

4. Applied egg-rr 99.7%

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

   1. clear-num N/A

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

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

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

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

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

   4. PI-lowering-PI.f64 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

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

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

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

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

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

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

   8. PI-lowering-PI.f64 99.7%

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

9. Simplified 99.7%

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

Alternative 2: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \pi}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.6e-77)
   (/ (/ (/ (* 0.5 PI) b) a) a)
   (/ (/ (/ (* 0.5 PI) a) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.6e-77) {
		tmp = (((0.5 * ((double) M_PI)) / b) / a) / a;
	} else {
		tmp = (((0.5 * ((double) M_PI)) / a) / b) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.6e-77) {
		tmp = (((0.5 * Math.PI) / b) / a) / a;
	} else {
		tmp = (((0.5 * Math.PI) / a) / b) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.6e-77:
		tmp = (((0.5 * math.pi) / b) / a) / a
	else:
		tmp = (((0.5 * math.pi) / a) / b) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.6e-77)
		tmp = Float64(Float64(Float64(Float64(0.5 * pi) / b) / a) / a);
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * pi) / a) / b) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.6e-77)
		tmp = (((0.5 * pi) / b) / a) / a;
	else
		tmp = (((0.5 * pi) / a) / b) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.6e-77], N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.5999999999999999e-77

   1. Initial program 77.0%

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

   3. 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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

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

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

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

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

      8. clear-num N/A

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

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

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

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

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

      11. /-lowering-/.f64 73.5%

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

   9. Applied egg-rr 73.5%

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

       1. associate-*l/ N/A

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

       2. frac-times N/A

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

       3. *-commutative N/A

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

       4. associate-/l/ N/A

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

       5. associate-*r/ N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

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

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

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

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

       11. PI-lowering-PI.f64 84.1%

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

   11. Applied egg-rr 84.1%

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

   if -5.5999999999999999e-77 < a

   1. Initial program 79.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. clear-num N/A

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

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

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

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

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

      4. PI-lowering-PI.f64 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

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

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

       9. associate-*r/ N/A

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

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

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

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

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

       12. PI-lowering-PI.f64 76.1%

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

   12. Simplified 76.1%

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

Alternative 3: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \pi}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.4e-77)
   (* (/ (/ PI a) b) (/ 0.5 a))
   (/ (/ (/ (* 0.5 PI) a) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.4e-77) {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / a);
	} else {
		tmp = (((0.5 * ((double) M_PI)) / a) / b) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.4e-77) {
		tmp = ((Math.PI / a) / b) * (0.5 / a);
	} else {
		tmp = (((0.5 * Math.PI) / a) / b) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.4e-77:
		tmp = ((math.pi / a) / b) * (0.5 / a)
	else:
		tmp = (((0.5 * math.pi) / a) / b) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.4e-77)
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * pi) / a) / b) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.4e-77)
		tmp = ((pi / a) / b) * (0.5 / a);
	else
		tmp = (((0.5 * pi) / a) / b) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.4e-77], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.39999999999999999e-77

   1. Initial program 77.0%

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

   3. 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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. clear-num N/A

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

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

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

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

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

      12. /-lowering-/.f64 84.3%

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

   9. Applied egg-rr 84.3%

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

   if -6.39999999999999999e-77 < a

   1. Initial program 79.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. clear-num N/A

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

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

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

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

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

      4. PI-lowering-PI.f64 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

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

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

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

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

       9. associate-*r/ N/A

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

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

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

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

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

       12. PI-lowering-PI.f64 76.1%

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

   12. Simplified 76.1%

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

Alternative 4: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b}{\frac{\pi}{a}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.4e-77)
   (* (/ (/ PI a) b) (/ 0.5 a))
   (/ (/ 0.5 b) (/ b (/ PI a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.4e-77) {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / a);
	} else {
		tmp = (0.5 / b) / (b / (((double) M_PI) / a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.4e-77) {
		tmp = ((Math.PI / a) / b) * (0.5 / a);
	} else {
		tmp = (0.5 / b) / (b / (Math.PI / a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.4e-77:
		tmp = ((math.pi / a) / b) * (0.5 / a)
	else:
		tmp = (0.5 / b) / (b / (math.pi / a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.4e-77)
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(b / Float64(pi / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.4e-77)
		tmp = ((pi / a) / b) * (0.5 / a);
	else
		tmp = (0.5 / b) / (b / (pi / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.4e-77], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(b / N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.39999999999999999e-77

   1. Initial program 77.0%

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

   3. 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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. clear-num N/A

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

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

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

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

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

      12. /-lowering-/.f64 84.3%

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

   9. Applied egg-rr 84.3%

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

   if -6.39999999999999999e-77 < a

   1. Initial program 79.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. div-inv N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.7%

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

   7. Simplified 67.7%

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. associate-/l/ N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

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

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

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

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. clear-num N/A

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

      15. associate-*l/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft-neg-in N/A

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

      18. neg-mul-1 N/A

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

      19. remove-double-neg N/A

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

      20. /-lowering-/.f64 N/A

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

      21. /-lowering-/.f64 N/A

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

      22. PI-lowering-PI.f64 76.1%

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

   9. Applied egg-rr 76.1%

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

Alternative 5: 74.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \frac{a}{\frac{0.5}{b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-78)
   (* (/ (/ PI a) b) (/ 0.5 a))
   (/ PI (* b (/ a (/ 0.5 b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-78) {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / a);
	} else {
		tmp = ((double) M_PI) / (b * (a / (0.5 / b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-78) {
		tmp = ((Math.PI / a) / b) * (0.5 / a);
	} else {
		tmp = Math.PI / (b * (a / (0.5 / b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.3e-78:
		tmp = ((math.pi / a) / b) * (0.5 / a)
	else:
		tmp = math.pi / (b * (a / (0.5 / b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.3e-78)
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / a));
	else
		tmp = Float64(pi / Float64(b * Float64(a / Float64(0.5 / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e-78)
		tmp = ((pi / a) / b) * (0.5 / a);
	else
		tmp = pi / (b * (a / (0.5 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.3e-78], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(b * N[(a / N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000002e-78

   1. Initial program 77.0%

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

   3. 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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. clear-num N/A

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

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

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

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

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

      12. /-lowering-/.f64 84.3%

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

   9. Applied egg-rr 84.3%

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

   if -2.3000000000000002e-78 < a

   1. Initial program 79.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. div-inv N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.7%

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

   7. Simplified 67.7%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

      13. /-lowering-/.f64 75.2%

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

   9. Applied egg-rr 75.2%

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

Alternative 6: 68.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.3e-77)
   (* (/ (/ PI a) b) (/ 0.5 a))
   (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.3e-77) {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / 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 <= -5.3e-77) {
		tmp = ((Math.PI / a) / b) * (0.5 / a);
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.3e-77:
		tmp = ((math.pi / a) / b) * (0.5 / a)
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.3e-77)
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / a));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.3e-77)
		tmp = ((pi / a) / b) * (0.5 / a);
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.3e-77], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.30000000000000015e-77

   1. Initial program 77.0%

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

   3. 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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. clear-num N/A

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

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

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

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

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

      12. /-lowering-/.f64 84.3%

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

   9. Applied egg-rr 84.3%

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

   if -5.30000000000000015e-77 < a

   1. Initial program 79.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. div-inv N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.7%

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

   7. Simplified 67.7%

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

Alternative 7: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-78}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-78)
   (* PI (/ (/ 0.5 a) (* b a)))
   (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-78) {
		tmp = ((double) M_PI) * ((0.5 / a) / (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-78) {
		tmp = Math.PI * ((0.5 / a) / (b * a));
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.8e-78:
		tmp = math.pi * ((0.5 / a) / (b * a))
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-78)
		tmp = Float64(pi * Float64(Float64(0.5 / a) / Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e-78)
		tmp = pi * ((0.5 / a) / (b * a));
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e-78], N[(Pi * N[(N[(0.5 / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.80000000000000023e-78

   1. Initial program 77.0%

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

   3. 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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. *-lowering-*.f64 84.2%

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

   9. Applied egg-rr 84.2%

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

   if -6.80000000000000023e-78 < a

   1. Initial program 79.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. div-inv N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.7%

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

   7. Simplified 67.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-78}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-77}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e-77)
   (* PI (/ 0.5 (* a (* b a))))
   (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e-77) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.5e-77) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.5e-77:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e-77)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.5e-77)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e-77], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.50000000000000008e-77

   1. Initial program 77.0%

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

   3. 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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. *-lowering-*.f64 82.8%

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

   7. Applied egg-rr 82.8%

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

   if -1.50000000000000008e-77 < a

   1. Initial program 79.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. div-inv N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.7%

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

   7. Simplified 67.7%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-77}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in b around 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

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

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

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 55.3%

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

5. Simplified 55.3%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. associate-*l* N/A

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

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

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

   8. *-lowering-*.f64 59.3%

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

7. Applied egg-rr 59.3%

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

8. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2024160 
(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))))