NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 14.7s

Alternatives: 10

Speedup: 1.3×

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. 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. associate-/l/ N/A

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

   9. div-inv N/A

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

   10. difference-of-squares N/A

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

   11. times-frac N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \left(\frac{a}{\pi} + \frac{b}{\pi}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e+156)
   (* (/ (/ PI b) a) (/ 0.5 (- a b)))
   (/ (/ 0.5 b) (* a (+ (/ a PI) (/ b PI))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+156) {
		tmp = ((((double) M_PI) / b) / a) * (0.5 / (a - b));
	} else {
		tmp = (0.5 / b) / (a * ((a / ((double) M_PI)) + (b / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+156) {
		tmp = ((Math.PI / b) / a) * (0.5 / (a - b));
	} else {
		tmp = (0.5 / b) / (a * ((a / Math.PI) + (b / Math.PI)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.5e+156:
		tmp = ((math.pi / b) / a) * (0.5 / (a - b))
	else:
		tmp = (0.5 / b) / (a * ((a / math.pi) + (b / math.pi)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e+156)
		tmp = Float64(Float64(Float64(pi / b) / a) * Float64(0.5 / Float64(a - b)));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(a * Float64(Float64(a / pi) + Float64(b / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e+156)
		tmp = ((pi / b) / a) * (0.5 / (a - b));
	else
		tmp = (0.5 / b) / (a * ((a / pi) + (b / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e+156], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[(0.5 / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(a * N[(N[(a / Pi), $MachinePrecision] + N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.49999999999999948e156

   1. Initial program 53.2%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. associate-/l/ N/A

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

      9. div-inv N/A

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

      10. difference-of-squares N/A

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

      11. times-frac N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 - \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right), \mathsf{/.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{\mathsf{PI}\left(\right)}{b \cdot a}\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{\frac{\mathsf{PI}\left(\right)}{b}}{a}\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. PI-lowering-PI.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 99.9%

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

   if -8.49999999999999948e156 < a

   1. Initial program 84.3%

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

      1. un-div-inv N/A

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

      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. associate-/l/ N/A

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

      9. div-inv N/A

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

      10. difference-of-squares N/A

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

      11. times-frac N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. clear-num N/A

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

      2. frac-times N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 90.5%

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

       1. associate-/r* N/A

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

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

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

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

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

       4. associate-/l* N/A

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

       5. associate-/l* N/A

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

       6. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

       12. PI-lowering-PI.f64 97.2%

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

   11. Applied egg-rr 97.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \left(\frac{a}{\pi} + \frac{b}{\pi}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot \left(\frac{a}{\pi} \cdot \left(a + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e+156)
   (* (/ (/ PI b) a) (/ 0.5 (- a b)))
   (/ 0.5 (* b (* (/ a PI) (+ a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+156) {
		tmp = ((((double) M_PI) / b) / a) * (0.5 / (a - b));
	} else {
		tmp = 0.5 / (b * ((a / ((double) M_PI)) * (a + b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+156) {
		tmp = ((Math.PI / b) / a) * (0.5 / (a - b));
	} else {
		tmp = 0.5 / (b * ((a / Math.PI) * (a + b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.5e+156:
		tmp = ((math.pi / b) / a) * (0.5 / (a - b))
	else:
		tmp = 0.5 / (b * ((a / math.pi) * (a + b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e+156)
		tmp = Float64(Float64(Float64(pi / b) / a) * Float64(0.5 / Float64(a - b)));
	else
		tmp = Float64(0.5 / Float64(b * Float64(Float64(a / pi) * Float64(a + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e+156)
		tmp = ((pi / b) / a) * (0.5 / (a - b));
	else
		tmp = 0.5 / (b * ((a / pi) * (a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e+156], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[(0.5 / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(b * N[(N[(a / Pi), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.49999999999999948e156

   1. Initial program 53.2%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. associate-/l/ N/A

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

      9. div-inv N/A

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

      10. difference-of-squares N/A

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

      11. times-frac N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 - \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right), \mathsf{/.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)\right), \mathsf{/.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{\mathsf{PI}\left(\right)}{b \cdot a}\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{\frac{\mathsf{PI}\left(\right)}{b}}{a}\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. PI-lowering-PI.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), a\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 99.9%

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

   if -8.49999999999999948e156 < a

   1. Initial program 84.3%

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

      1. un-div-inv N/A

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

      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. associate-/l/ N/A

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

      9. div-inv N/A

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

      10. difference-of-squares N/A

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

      11. times-frac N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. clear-num N/A

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

      2. frac-times N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 90.5%

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

   10. Taylor expanded in a around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*l/ N/A

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

       3. *-commutative N/A

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

       4. associate-*l/ N/A

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

       5. distribute-lft-out N/A

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

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

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

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

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

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

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

       9. +-lowering-+.f64 96.2%

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

   12. Simplified 96.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot \left(\frac{a}{\pi} \cdot \left(a + b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot b}}{\frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot \left(\frac{a}{\pi} \cdot \left(a + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.25e+94)
   (/ (/ 0.5 (* a b)) (/ a PI))
   (/ 0.5 (* b (* (/ a PI) (+ a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.25e+94) {
		tmp = (0.5 / (a * b)) / (a / ((double) M_PI));
	} else {
		tmp = 0.5 / (b * ((a / ((double) M_PI)) * (a + b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.25e+94) {
		tmp = (0.5 / (a * b)) / (a / Math.PI);
	} else {
		tmp = 0.5 / (b * ((a / Math.PI) * (a + b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.25e+94:
		tmp = (0.5 / (a * b)) / (a / math.pi)
	else:
		tmp = 0.5 / (b * ((a / math.pi) * (a + b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.25e+94)
		tmp = Float64(Float64(0.5 / Float64(a * b)) / Float64(a / pi));
	else
		tmp = Float64(0.5 / Float64(b * Float64(Float64(a / pi) * Float64(a + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.25e+94)
		tmp = (0.5 / (a * b)) / (a / pi);
	else
		tmp = 0.5 / (b * ((a / pi) * (a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.25e+94], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a / Pi), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(b * N[(N[(a / Pi), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.25000000000000003e94

   1. Initial program 62.4%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

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

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

      8. associate-*r/ N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 87.5%

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

   5. Simplified 87.5%

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

      1. associate-/l/ N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r/ N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

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

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. PI-lowering-PI.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.25000000000000003e94 < a

   1. Initial program 83.7%

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

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

      9. div-inv N/A

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

      10. difference-of-squares N/A

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

      11. times-frac N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. clear-num N/A

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

      2. frac-times N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 90.1%

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

   10. Taylor expanded in a around 0

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

       1. distribute-rgt-in N/A

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

       2. associate-*l/ N/A

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

       3. *-commutative N/A

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

       4. associate-*l/ N/A

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

       5. distribute-lft-out N/A

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

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

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

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

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

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

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

       9. +-lowering-+.f64 96.1%

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

   12. Simplified 96.1%

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

Alternative 5: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(\frac{b}{\pi} \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.05e+80)
   (/ 0.5 (* a (* (/ b PI) (+ a b))))
   (/ (/ (/ PI b) a) (/ b 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.05e+80) {
		tmp = 0.5 / (a * ((b / ((double) M_PI)) * (a + b)));
	} else {
		tmp = ((((double) M_PI) / b) / a) / (b / 0.5);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.05e+80) {
		tmp = 0.5 / (a * ((b / Math.PI) * (a + b)));
	} else {
		tmp = ((Math.PI / b) / a) / (b / 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.05e+80:
		tmp = 0.5 / (a * ((b / math.pi) * (a + b)))
	else:
		tmp = ((math.pi / b) / a) / (b / 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.05e+80)
		tmp = Float64(0.5 / Float64(a * Float64(Float64(b / pi) * Float64(a + b))));
	else
		tmp = Float64(Float64(Float64(pi / b) / a) / Float64(b / 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.05e+80)
		tmp = 0.5 / (a * ((b / pi) * (a + b)));
	else
		tmp = ((pi / b) / a) / (b / 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.05e+80], N[(0.5 / N[(a * N[(N[(b / Pi), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] / N[(b / 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05000000000000001e80

   1. Initial program 82.7%

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

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

      9. div-inv N/A

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

      10. difference-of-squares N/A

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

      11. times-frac N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. clear-num N/A

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

      2. frac-times N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 86.9%

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

   10. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l/ N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. associate-*r/ N/A

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

       7. associate-*l/ N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. associate-/l* N/A

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

       11. distribute-lft-in N/A

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

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

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

       13. associate-/l* N/A

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

       14. unpow2 N/A

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

       15. associate-/l* N/A

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

       16. distribute-rgt-out N/A

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

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

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

   12. Simplified 95.0%

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

   if 1.05000000000000001e80 < b

   1. Initial program 72.7%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 82.7%

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

   5. Simplified 82.7%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 99.9%

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

   9. Applied egg-rr 99.9%

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

Alternative 6: 77.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\pi \cdot \frac{\frac{-0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-77)
   (* PI (/ (/ -0.5 (* a b)) (- b a)))
   (/ (/ (/ PI b) a) (/ b 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-77) {
		tmp = ((double) M_PI) * ((-0.5 / (a * b)) / (b - a));
	} else {
		tmp = ((((double) M_PI) / b) / a) / (b / 0.5);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-77) {
		tmp = Math.PI * ((-0.5 / (a * b)) / (b - a));
	} else {
		tmp = ((Math.PI / b) / a) / (b / 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.5e-77:
		tmp = math.pi * ((-0.5 / (a * b)) / (b - a))
	else:
		tmp = ((math.pi / b) / a) / (b / 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-77)
		tmp = Float64(pi * Float64(Float64(-0.5 / Float64(a * b)) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi / b) / a) / Float64(b / 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-77)
		tmp = pi * ((-0.5 / (a * b)) / (b - a));
	else
		tmp = ((pi / b) / a) / (b / 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.5e-77], N[(Pi * N[(N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] / N[(b / 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5000000000000001e-77

   1. Initial program 78.7%

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

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in a around inf

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. /-lowering-/.f64 86.0%

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

   7. Simplified 86.0%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. --lowering--.f64 85.9%

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

   9. Applied egg-rr 85.9%

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

   if -4.5000000000000001e-77 < a

   1. Initial program 81.3%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 61.4%

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

   5. Simplified 61.4%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 70.9%

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

   9. Applied egg-rr 70.9%

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

Alternative 7: 75.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6e-16) (* (/ 0.5 a) (/ PI (* a b))) (/ (/ (/ PI b) a) (/ b 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-16) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = ((((double) M_PI) / b) / a) / (b / 0.5);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-16) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = ((Math.PI / b) / a) / (b / 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.6e-16:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = ((math.pi / b) / a) / (b / 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6e-16)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(pi / b) / a) / Float64(b / 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.6e-16)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = ((pi / b) / a) / (b / 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6e-16], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] / N[(b / 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.60000000000000011e-16

   1. Initial program 81.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

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

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

      8. associate-*r/ N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 66.0%

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

   5. Simplified 66.0%

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

      1. associate-/l/ N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 71.8%

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

   9. Applied egg-rr 71.8%

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

   if 1.60000000000000011e-16 < b

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 80.9%

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

   5. Simplified 80.9%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 94.2%

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

   9. Applied egg-rr 94.2%

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

Alternative 8: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-17}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22e-17)
   (* (/ 0.5 a) (/ PI (* a b)))
   (* (/ 0.5 b) (/ (/ PI a) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22e-17) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.22e-17) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = (0.5 / b) * ((Math.PI / a) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.22e-17:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = (0.5 / b) * ((math.pi / a) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.22e-17)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.22e-17)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = (0.5 / b) * ((pi / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.22e-17], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22e-17

   1. Initial program 81.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

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

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

      8. associate-*r/ N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 66.0%

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

   5. Simplified 66.0%

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

      1. associate-/l/ N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 71.8%

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

   9. Applied egg-rr 71.8%

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

   if 1.22e-17 < b

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 80.9%

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

   5. Simplified 80.9%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 94.2%

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

   7. Applied egg-rr 94.2%

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

Alternative 9: 62.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in b around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

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

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

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

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

   8. associate-*r/ N/A

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

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

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

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

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

   11. PI-lowering-PI.f64 56.6%

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

5. Simplified 56.6%

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

   1. associate-/l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 60.6%

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

7. Applied egg-rr 60.6%

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

   1. associate-*r/ N/A

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

   2. times-frac N/A

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

   3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 60.6%

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

9. Applied egg-rr 60.6%

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

Alternative 10: 62.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Taylor expanded in b around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

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

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

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

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

   8. associate-*r/ N/A

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

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

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

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

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

   11. PI-lowering-PI.f64 56.6%

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

5. Simplified 56.6%

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

   1. associate-/l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 60.6%

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

7. Applied egg-rr 60.6%

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

Reproduce

herbie shell --seed 2024192 
(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))))