NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 12.4s

Alternatives: 6

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. difference-of-squares N/A

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

   5. times-frac N/A

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.5%

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

   1. associate-*l* N/A

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

   2. frac-times N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

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

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

6. Applied egg-rr 99.7%

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

Alternative 2: 74.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.15e-44)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.15e-44) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.15e-44) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.15e-44:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.15e-44)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.15e-44)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.15e-44], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.15000000000000007e-44

   1. Initial program 89.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

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

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

      3. +-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 89.1

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

   5. Simplified 89.1%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. PI-lowering-PI.f64 89.1

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

   7. Applied egg-rr 89.1%

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

   if -2.15000000000000007e-44 < a

   1. Initial program 77.9%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

      1. associate-*l* N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 70.4

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

   9. Simplified 70.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. div-inv N/A

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

       5. frac-times N/A

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

       6. metadata-eval N/A

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

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

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

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

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

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

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

       10. PI-lowering-PI.f64 70.5

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

   11. Applied egg-rr 70.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-44}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3e-44) (* PI (/ 0.5 (* a (* b a)))) (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3e-44) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3e-44) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3e-44:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3e-44)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3e-44)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3e-44], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.0000000000000002e-44

   1. Initial program 89.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

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

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

      3. +-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 89.1

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

   5. Simplified 89.1%

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

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 83.2

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

   7. Applied egg-rr 83.2%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. PI-lowering-PI.f64 83.1

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

   9. Applied egg-rr 83.1%

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

       1. associate-*r* N/A

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

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

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

       3. *-lowering-*.f64 89.0

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

   11. Applied egg-rr 89.0%

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

   if -3.0000000000000002e-44 < a

   1. Initial program 77.9%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

      1. associate-*l* N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 70.4

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

   9. Simplified 70.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. div-inv N/A

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

       5. frac-times N/A

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

       6. metadata-eval N/A

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

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

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

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

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

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

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

       10. PI-lowering-PI.f64 70.5

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

   11. Applied egg-rr 70.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-44}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.15e-44)
   (* 0.5 (/ PI (* b (* a a))))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.15e-44) {
		tmp = 0.5 * (((double) M_PI) / (b * (a * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.15e-44) {
		tmp = 0.5 * (Math.PI / (b * (a * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.15e-44:
		tmp = 0.5 * (math.pi / (b * (a * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.15e-44)
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.15e-44)
		tmp = 0.5 * (pi / (b * (a * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.15e-44], N[(0.5 * N[(Pi / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.15000000000000007e-44

   1. Initial program 89.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

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

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

      3. +-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 89.1

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

   5. Simplified 89.1%

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

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 83.2

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

   7. Applied egg-rr 83.2%

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

   if -2.15000000000000007e-44 < a

   1. Initial program 77.9%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

      1. associate-*l* N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 70.4

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

   9. Simplified 70.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. div-inv N/A

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

       5. frac-times N/A

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

       6. metadata-eval N/A

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

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

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

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

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

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

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

       10. PI-lowering-PI.f64 70.5

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

   11. Applied egg-rr 70.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. difference-of-squares N/A

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

   5. times-frac N/A

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.5%

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

   1. associate-*l* N/A

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

   2. frac-times N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

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

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/l/ N/A

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

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

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

   6. /-rgt-identity N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. remove-double-neg N/A

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

   10. neg-mul-1 N/A

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

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

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

   12. metadata-eval N/A

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

   13. associate-*l/ N/A

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

   14. clear-num N/A

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

   15. /-rgt-identity N/A

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

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

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

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

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

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

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

   19. PI-lowering-PI.f64 99.0

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 6: 56.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

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

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

   3. +-rgt-identity N/A

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

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

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

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

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-lowering-*.f64 64.6

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

5. Simplified 64.6%

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

   1. +-rgt-identity N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 60.1

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

7. Applied egg-rr 60.1%

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

Reproduce

herbie shell --seed 2024196 
(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))))