NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 11.2s

Alternatives: 9

Speedup: 2.0×

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: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

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

   13. *-lft-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

6. Applied rewrites 99.7%

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

Alternative 2: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(2 \cdot \left(b + a\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+151)
   (/ (/ PI a) (* 2.0 (* b a)))
   (/ PI (* b (* a (* 2.0 (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+151) {
		tmp = (((double) M_PI) / a) / (2.0 * (b * a));
	} else {
		tmp = ((double) M_PI) / (b * (a * (2.0 * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e+151) {
		tmp = (Math.PI / a) / (2.0 * (b * a));
	} else {
		tmp = Math.PI / (b * (a * (2.0 * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e+151:
		tmp = (math.pi / a) / (2.0 * (b * a))
	else:
		tmp = math.pi / (b * (a * (2.0 * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+151)
		tmp = Float64(Float64(pi / a) / Float64(2.0 * Float64(b * a)));
	else
		tmp = Float64(pi / Float64(b * Float64(a * Float64(2.0 * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+151)
		tmp = (pi / a) / (2.0 * (b * a));
	else
		tmp = pi / (b * (a * (2.0 * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+151], N[(N[(Pi / a), $MachinePrecision] / N[(2.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(b * N[(a * N[(2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000003e151

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -2.00000000000000003e151 < 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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. frac-times N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(2 \cdot \left(b + a\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(2 \cdot \left(b + a\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.6e+151)
   (/ (* PI (/ 0.5 a)) (* b a))
   (/ PI (* b (* a (* 2.0 (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.6e+151) {
		tmp = (((double) M_PI) * (0.5 / a)) / (b * a);
	} else {
		tmp = ((double) M_PI) / (b * (a * (2.0 * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.6e+151) {
		tmp = (Math.PI * (0.5 / a)) / (b * a);
	} else {
		tmp = Math.PI / (b * (a * (2.0 * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.6e+151:
		tmp = (math.pi * (0.5 / a)) / (b * a)
	else:
		tmp = math.pi / (b * (a * (2.0 * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.6e+151)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(b * a));
	else
		tmp = Float64(pi / Float64(b * Float64(a * Float64(2.0 * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.6e+151)
		tmp = (pi * (0.5 / a)) / (b * a);
	else
		tmp = pi / (b * (a * (2.0 * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.6e+151], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(b * N[(a * N[(2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.59999999999999975e151

   1. Initial program 54.6%

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

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 99.8%

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

   if -5.59999999999999975e151 < 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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. frac-times N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(2 \cdot \left(b + a\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+152}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(2 \cdot \left(b + a\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.45e+152)
   (* PI (/ 0.5 (* a (* b a))))
   (/ PI (* b (* a (* 2.0 (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.45e+152) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) / (b * (a * (2.0 * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.45e+152) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI / (b * (a * (2.0 * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.45e+152:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi / (b * (a * (2.0 * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.45e+152)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi / Float64(b * Float64(a * Float64(2.0 * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.45e+152)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi / (b * (a * (2.0 * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.45e+152], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(b * N[(a * N[(2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4499999999999999e152

   1. Initial program 54.6%

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

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

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

   if -1.4499999999999999e152 < 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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. frac-times N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+152}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(2 \cdot \left(b + a\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+113)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+113) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+113) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+113:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = math.pi * (0.5 / (b * (a * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+113)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+113)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = pi * (0.5 / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+113], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5e113

   1. Initial program 62.5%

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

   3. 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -5e113 < a

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. *-rgt-identity N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 99.2

          \[\leadsto \left(\pi \cdot 0.5\right) \cdot \frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b \cdot a\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\pi \cdot 0.5\right) \cdot \frac{1}{\left(b + a\right) \cdot \left(b \cdot a\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 99.2

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-+.f64 96.6

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

   10. Applied rewrites 96.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(b + a\right) \cdot \left(b \cdot a\right)} \cdot \left(\pi \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ 1.0 (* (+ b a) (* b a))) (* PI 0.5)))

C

double code(double a, double b) {
	return (1.0 / ((b + a) * (b * a))) * (((double) M_PI) * 0.5);
}

Java

public static double code(double a, double b) {
	return (1.0 / ((b + a) * (b * a))) * (Math.PI * 0.5);
}

Python

def code(a, b):
	return (1.0 / ((b + a) * (b * a))) * (math.pi * 0.5)

Julia

function code(a, b)
	return Float64(Float64(1.0 / Float64(Float64(b + a) * Float64(b * a))) * Float64(pi * 0.5))
end

MATLAB

function tmp = code(a, b)
	tmp = (1.0 / ((b + a) * (b * a))) * (pi * 0.5);
end

Wolfram

code[a_, b_] := N[(N[(1.0 / N[(N[(b + a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

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

   13. *-lft-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. frac-times N/A

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

   10. *-rgt-identity N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 99.1

       \[\leadsto \left(\pi \cdot 0.5\right) \cdot \frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b \cdot a\right)}} \]

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\pi \cdot 0.5\right) \cdot \frac{1}{\left(b + a\right) \cdot \left(b \cdot a\right)}} \]

9. Final simplification 99.1%

   \[\leadsto \frac{1}{\left(b + a\right) \cdot \left(b \cdot a\right)} \cdot \left(\pi \cdot 0.5\right) \]
10. Add Preprocessing

Alternative 7: 71.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-95) (/ (* PI 0.5) (* a (* b a))) (/ (* PI 0.5) (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-95) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = (((double) M_PI) * 0.5) / (a * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-95) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (Math.PI * 0.5) / (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.1e-95:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (math.pi * 0.5) / (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e-95)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.1e-95)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (pi * 0.5) / (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e-95], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0999999999999999e-95

   1. Initial program 78.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if 1.0999999999999999e-95 < b

   1. Initial program 85.9%

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

   3. Taylor expanded in b around 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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

3. 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 64.0

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

5. Applied rewrites 64.0%

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

6. Final simplification 64.0%

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

Alternative 9: 63.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

3. 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 64.0

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

5. Applied rewrites 64.0%

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

7. Applied rewrites 63.9%

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

Reproduce

herbie shell --seed 2024216 
(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))))