NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.2% → 99.7%

Time: 11.7s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. *-commutative 78.1%

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

   2. associate-*r* 78.1%

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

   3. associate-*r/ 78.1%

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

   4. associate-*r* 78.1%

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

   5. *-rgt-identity 78.1%

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

   6. sub-neg 78.1%

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

   7. distribute-neg-frac 78.1%

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

   8. metadata-eval 78.1%

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

3. Simplified 78.1%

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

   1. *-commutative 78.1%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.3%

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

   4. div-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. add-sqr-sqrt 50.2%

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

   7. sqrt-unprod 72.4%

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

   8. frac-times 72.4%

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

   9. metadata-eval 72.4%

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

   10. metadata-eval 72.4%

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

   11. frac-times 72.4%

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

   12. sqrt-unprod 31.8%

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

   13. add-sqr-sqrt 71.5%

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

6. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
7. Step-by-step derivation

   1. *-commutative 71.5%

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

   2. +-commutative 71.5%

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

   3. associate-/l* 71.5%

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

   4. +-commutative 71.5%

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

8. Simplified 71.5%

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

9. Taylor expanded in b around inf 99.6%

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

    1. associate-*r/ 99.7%

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

    2. +-commutative 99.7%

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

11. Applied egg-rr 99.7%

    \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\frac{\pi \cdot 0.5}{b + a}} \]
12. Step-by-step derivation

    1. associate-*l/ 99.8%

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

    2. *-un-lft-identity 99.8%

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

    3. +-commutative 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot -0.5}{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.25e-27)
   (/ (* (/ PI a) -0.5) (* a (- b)))
   (* (/ 1.0 (* a b)) (* PI (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.25e-27) {
		tmp = ((((double) M_PI) / a) * -0.5) / (a * -b);
	} else {
		tmp = (1.0 / (a * b)) * (((double) M_PI) * (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.25e-27) {
		tmp = ((Math.PI / a) * -0.5) / (a * -b);
	} else {
		tmp = (1.0 / (a * b)) * (Math.PI * (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.25e-27:
		tmp = ((math.pi / a) * -0.5) / (a * -b)
	else:
		tmp = (1.0 / (a * b)) * (math.pi * (0.5 / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.25e-27)
		tmp = Float64(Float64(Float64(pi / a) * -0.5) / Float64(a * Float64(-b)));
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(pi * Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.25e-27)
		tmp = ((pi / a) * -0.5) / (a * -b);
	else
		tmp = (1.0 / (a * b)) * (pi * (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.25e-27], N[(N[(N[(Pi / a), $MachinePrecision] * -0.5), $MachinePrecision] / N[(a * (-b)), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2500000000000001e-27

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*r* 79.8%

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

      3. associate-*r/ 79.8%

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

      4. associate-*r* 79.8%

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

      5. *-rgt-identity 79.8%

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

      6. sub-neg 79.8%

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

      7. distribute-neg-frac 79.8%

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

      8. metadata-eval 79.8%

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

   3. Simplified 79.8%

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

      1. *-commutative 79.8%

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

      2. difference-of-squares 88.2%

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

      3. times-frac 99.2%

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

      4. div-inv 99.2%

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

      5. metadata-eval 99.2%

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

      6. add-sqr-sqrt 67.3%

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

      7. sqrt-unprod 65.1%

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

      8. frac-times 65.1%

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

      9. metadata-eval 65.1%

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

      10. metadata-eval 65.1%

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

      11. frac-times 65.1%

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

      12. sqrt-unprod 10.7%

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

      13. add-sqr-sqrt 63.9%

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

   6. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
   7. Step-by-step derivation

      1. *-commutative 63.9%

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

      2. +-commutative 63.9%

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

      3. associate-/l* 63.9%

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

      4. +-commutative 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in b around inf 99.6%

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

   10. Taylor expanded in a around inf 68.1%

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

       1. associate-*l/ 68.2%

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

       2. *-un-lft-identity 68.2%

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

       3. associate-*r/ 68.2%

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

       4. *-commutative 68.2%

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

       5. associate-*r/ 68.2%

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

       6. frac-2neg 68.2%

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

       7. *-commutative 68.2%

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

       8. distribute-rgt-neg-in 68.2%

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

       9. metadata-eval 68.2%

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

       10. distribute-rgt-neg-in 68.2%

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

   12. Applied egg-rr 68.2%

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

   if 2.2500000000000001e-27 < b

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. associate-*r* 73.3%

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

      3. associate-*r/ 73.3%

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

      4. associate-*r* 73.3%

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

      5. *-rgt-identity 73.3%

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

      6. sub-neg 73.3%

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

      7. distribute-neg-frac 73.3%

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

      8. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

      1. *-commutative 73.3%

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

      2. difference-of-squares 88.7%

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

      3. times-frac 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 93.9%

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

      8. frac-times 93.9%

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

      9. metadata-eval 93.9%

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

      10. metadata-eval 93.9%

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

      11. frac-times 93.9%

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

      12. sqrt-unprod 93.9%

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

      13. add-sqr-sqrt 93.9%

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

   6. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
   7. Step-by-step derivation

      1. *-commutative 93.9%

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

      2. +-commutative 93.9%

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

      3. associate-/l* 94.0%

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

      4. +-commutative 94.0%

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

   8. Simplified 94.0%

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

   9. Taylor expanded in b around inf 99.6%

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

   10. Taylor expanded in a around 0 84.0%

       \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \color{blue}{\frac{0.5}{b}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot -0.5}{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e+142)
   (* (/ PI a) (/ (/ 0.5 b) (+ a b)))
   (* (/ 1.0 (* a b)) (* PI (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e+142) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / (a + b));
	} else {
		tmp = (1.0 / (a * b)) * (((double) M_PI) * (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2e+142) {
		tmp = (Math.PI / a) * ((0.5 / b) / (a + b));
	} else {
		tmp = (1.0 / (a * b)) * (Math.PI * (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2e+142:
		tmp = (math.pi / a) * ((0.5 / b) / (a + b))
	else:
		tmp = (1.0 / (a * b)) * (math.pi * (0.5 / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e+142)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / Float64(a + b)));
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(pi * Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2e+142)
		tmp = (pi / a) * ((0.5 / b) / (a + b));
	else
		tmp = (1.0 / (a * b)) * (pi * (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e+142], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.0000000000000001e142

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*r* 82.9%

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

      3. associate-*r/ 82.9%

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

      4. associate-*r* 82.9%

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

      5. *-rgt-identity 82.9%

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

      6. sub-neg 82.9%

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

      7. distribute-neg-frac 82.9%

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

      8. metadata-eval 82.9%

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

   3. Simplified 82.9%

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

      1. *-commutative 82.9%

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

      2. difference-of-squares 90.0%

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

      3. times-frac 99.2%

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

      4. div-inv 99.2%

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

      5. metadata-eval 99.2%

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

      6. add-sqr-sqrt 56.6%

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

      7. sqrt-unprod 68.9%

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

      8. frac-times 68.9%

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

      9. metadata-eval 68.9%

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

      10. metadata-eval 68.9%

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

      11. frac-times 68.9%

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

      12. sqrt-unprod 23.2%

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

      13. add-sqr-sqrt 67.9%

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

   6. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
   7. Step-by-step derivation

      1. *-commutative 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-/l* 67.9%

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

      4. +-commutative 67.9%

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

   8. Simplified 67.9%

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

   9. Taylor expanded in b around inf 99.6%

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

       1. associate-*r/ 99.7%

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

       2. +-commutative 99.7%

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

   11. Applied egg-rr 99.7%

       \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\frac{\pi \cdot 0.5}{b + a}} \]
   12. Step-by-step derivation

       1. clear-num 99.6%

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

       2. frac-2neg 99.6%

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

       3. metadata-eval 99.6%

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

       4. frac-2neg 99.6%

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

       5. metadata-eval 99.6%

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

       6. frac-times 99.4%

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

       7. metadata-eval 99.4%

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

       8. distribute-rgt-neg-in 99.4%

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

       9. distribute-neg-frac2 99.4%

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

       10. +-commutative 99.4%

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

       11. distribute-rgt-neg-in 99.4%

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

       12. metadata-eval 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. distribute-rgt-neg-out 99.4%

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

       2. distribute-lft-neg-out 99.4%

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

       3. associate-*r/ 99.4%

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

       4. associate-*r* 96.6%

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

       5. *-commutative 96.6%

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

       6. distribute-frac-neg2 96.6%

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

       7. *-commutative 96.6%

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

       8. distribute-rgt-neg-in 96.6%

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

       9. metadata-eval 96.6%

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

       10. associate-/r/ 96.6%

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

       11. associate-*l/ 96.6%

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

       12. *-lft-identity 96.6%

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

       13. times-frac 96.8%

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

       14. associate-/r* 96.3%

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

   15. Simplified 96.3%

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

   if 2.0000000000000001e142 < b

   1. Initial program 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-*r* 40.7%

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

      3. associate-*r/ 40.7%

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

      4. associate-*r* 40.7%

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

      5. *-rgt-identity 40.7%

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

      6. sub-neg 40.7%

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

      7. distribute-neg-frac 40.7%

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

      8. metadata-eval 40.7%

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

   3. Simplified 40.7%

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

      1. *-commutative 40.7%

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

      2. difference-of-squares 75.2%

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

      3. times-frac 99.8%

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

      4. div-inv 99.8%

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

      5. metadata-eval 99.8%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 99.8%

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

      8. frac-times 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. frac-times 99.8%

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

      12. sqrt-unprod 99.8%

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

      13. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
   7. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/l* 99.9%

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

      4. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in b around inf 99.8%

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

   10. Taylor expanded in a around 0 99.8%

       \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \color{blue}{\frac{0.5}{b}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+76}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e+76)
   (* (/ PI a) (/ (/ 0.5 b) (+ a b)))
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+76) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / (a + b));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+76) {
		tmp = (Math.PI / a) * ((0.5 / b) / (a + b));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.15e+76:
		tmp = (math.pi / a) * ((0.5 / b) / (a + b))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15e+76)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / Float64(a + b)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e+76)
		tmp = (pi / a) * ((0.5 / b) / (a + b));
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15e+76], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15000000000000001e76

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-*r* 81.9%

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

      3. associate-*r/ 81.9%

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

      4. associate-*r* 81.9%

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

      5. *-rgt-identity 81.9%

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

      6. sub-neg 81.9%

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

      7. distribute-neg-frac 81.9%

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

      8. metadata-eval 81.9%

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

   3. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. difference-of-squares 89.4%

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

      3. times-frac 99.2%

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

      4. div-inv 99.2%

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

      5. metadata-eval 99.2%

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

      6. add-sqr-sqrt 60.1%

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

      7. sqrt-unprod 67.4%

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

      8. frac-times 67.4%

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

      9. metadata-eval 67.4%

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

      10. metadata-eval 67.4%

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

      11. frac-times 67.4%

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

      12. sqrt-unprod 18.9%

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

      13. add-sqr-sqrt 66.4%

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

   6. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
   7. Step-by-step derivation

      1. *-commutative 66.4%

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

      2. +-commutative 66.4%

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

      3. associate-/l* 66.4%

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

      4. +-commutative 66.4%

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

   8. Simplified 66.4%

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

   9. Taylor expanded in b around inf 99.6%

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

       1. associate-*r/ 99.7%

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

       2. +-commutative 99.7%

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

   11. Applied egg-rr 99.7%

       \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\frac{\pi \cdot 0.5}{b + a}} \]
   12. Step-by-step derivation

       1. clear-num 99.6%

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

       2. frac-2neg 99.6%

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

       3. metadata-eval 99.6%

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

       4. frac-2neg 99.6%

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

       5. metadata-eval 99.6%

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

       6. frac-times 99.4%

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

       7. metadata-eval 99.4%

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

       8. distribute-rgt-neg-in 99.4%

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

       9. distribute-neg-frac2 99.4%

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

       10. +-commutative 99.4%

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

       11. distribute-rgt-neg-in 99.4%

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

       12. metadata-eval 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. distribute-rgt-neg-out 99.4%

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

       2. distribute-lft-neg-out 99.4%

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

       3. associate-*r/ 99.4%

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

       4. associate-*r* 96.4%

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

       5. *-commutative 96.4%

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

       6. distribute-frac-neg2 96.4%

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

       7. *-commutative 96.4%

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

       8. distribute-rgt-neg-in 96.4%

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

       9. metadata-eval 96.4%

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

       10. associate-/r/ 96.4%

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

       11. associate-*l/ 96.4%

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

       12. *-lft-identity 96.4%

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

       13. times-frac 96.6%

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

       14. associate-/r* 96.2%

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

   15. Simplified 96.2%

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

   if 1.15000000000000001e76 < b

   1. Initial program 58.9%

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

      1. associate-*l* 58.9%

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

      2. *-rgt-identity 58.9%

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

      3. associate-/l* 58.9%

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

      4. metadata-eval 58.9%

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

      5. associate-*l/ 58.9%

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

      6. *-lft-identity 58.9%

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

      7. sub-neg 58.9%

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

      8. distribute-neg-frac 58.9%

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

      9. metadata-eval 58.9%

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

   3. Simplified 58.9%

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

      1. metadata-eval 58.9%

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

      2. div-inv 58.9%

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

      3. associate-*r/ 58.9%

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

      4. *-commutative 58.9%

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

      5. difference-of-squares 82.7%

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

      6. associate-/r* 99.7%

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

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in a around 0 97.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+76}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. *-commutative 78.1%

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

   2. associate-*r* 78.1%

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

   3. associate-*r/ 78.1%

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

   4. associate-*r* 78.1%

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

   5. *-rgt-identity 78.1%

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

   6. sub-neg 78.1%

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

   7. distribute-neg-frac 78.1%

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

   8. metadata-eval 78.1%

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

3. Simplified 78.1%

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

   1. *-commutative 78.1%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.3%

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

   4. div-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. add-sqr-sqrt 50.2%

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

   7. sqrt-unprod 72.4%

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

   8. frac-times 72.4%

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

   9. metadata-eval 72.4%

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

   10. metadata-eval 72.4%

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

   11. frac-times 72.4%

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

   12. sqrt-unprod 31.8%

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

   13. add-sqr-sqrt 71.5%

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

6. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
7. Step-by-step derivation

   1. *-commutative 71.5%

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

   2. +-commutative 71.5%

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

   3. associate-/l* 71.5%

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

   4. +-commutative 71.5%

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

8. Simplified 71.5%

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

9. Taylor expanded in b around inf 99.6%

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

    1. associate-*l/ 99.7%

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

    2. *-un-lft-identity 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 6: 62.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. *-commutative 78.1%

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

   2. associate-*r* 78.1%

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

   3. associate-*r/ 78.1%

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

   4. associate-*r* 78.1%

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

   5. *-rgt-identity 78.1%

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

   6. sub-neg 78.1%

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

   7. distribute-neg-frac 78.1%

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

   8. metadata-eval 78.1%

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

3. Simplified 78.1%

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

   1. *-commutative 78.1%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.3%

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

   4. div-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. add-sqr-sqrt 50.2%

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

   7. sqrt-unprod 72.4%

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

   8. frac-times 72.4%

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

   9. metadata-eval 72.4%

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

   10. metadata-eval 72.4%

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

   11. frac-times 72.4%

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

   12. sqrt-unprod 31.8%

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

   13. add-sqr-sqrt 71.5%

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

6. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
7. Step-by-step derivation

   1. *-commutative 71.5%

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

   2. +-commutative 71.5%

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

   3. associate-/l* 71.5%

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

   4. +-commutative 71.5%

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

8. Simplified 71.5%

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

9. Taylor expanded in b around inf 99.6%

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

10. Taylor expanded in a around inf 64.8%

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

    1. associate-*l/ 64.9%

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

    2. *-un-lft-identity 64.9%

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

    3. associate-*r/ 64.9%

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

    4. *-commutative 64.9%

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

    5. associate-*r/ 64.9%

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

    6. frac-2neg 64.9%

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

    7. *-commutative 64.9%

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

    8. distribute-rgt-neg-in 64.9%

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

    9. metadata-eval 64.9%

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

    10. distribute-rgt-neg-in 64.9%

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

12. Applied egg-rr 64.9%

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

13. Final simplification 64.9%

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

Alternative 7: 62.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. *-commutative 78.1%

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

   2. associate-*r* 78.1%

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

   3. associate-*r/ 78.1%

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

   4. associate-*r* 78.1%

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

   5. *-rgt-identity 78.1%

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

   6. sub-neg 78.1%

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

   7. distribute-neg-frac 78.1%

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

   8. metadata-eval 78.1%

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

3. Simplified 78.1%

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

   1. *-commutative 78.1%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.3%

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

   4. div-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. add-sqr-sqrt 50.2%

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

   7. sqrt-unprod 72.4%

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

   8. frac-times 72.4%

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

   9. metadata-eval 72.4%

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

   10. metadata-eval 72.4%

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

   11. frac-times 72.4%

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

   12. sqrt-unprod 31.8%

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

   13. add-sqr-sqrt 71.5%

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

6. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
7. Step-by-step derivation

   1. *-commutative 71.5%

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

   2. +-commutative 71.5%

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

   3. associate-/l* 71.5%

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

   4. +-commutative 71.5%

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

8. Simplified 71.5%

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

9. Taylor expanded in b around inf 99.6%

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

10. Taylor expanded in a around inf 64.8%

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

    1. associate-*r/ 64.9%

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

    2. frac-times 64.7%

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

    3. *-un-lft-identity 64.7%

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

12. Applied egg-rr 64.7%

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

13. Final simplification 64.7%

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

Alternative 8: 62.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. *-commutative 78.1%

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

   2. associate-*r* 78.1%

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

   3. associate-*r/ 78.1%

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

   4. associate-*r* 78.1%

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

   5. *-rgt-identity 78.1%

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

   6. sub-neg 78.1%

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

   7. distribute-neg-frac 78.1%

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

   8. metadata-eval 78.1%

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

3. Simplified 78.1%

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

   1. *-commutative 78.1%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.3%

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

   4. div-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. add-sqr-sqrt 50.2%

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

   7. sqrt-unprod 72.4%

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

   8. frac-times 72.4%

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

   9. metadata-eval 72.4%

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

   10. metadata-eval 72.4%

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

   11. frac-times 72.4%

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

   12. sqrt-unprod 31.8%

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

   13. add-sqr-sqrt 71.5%

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

6. Applied egg-rr 71.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{\frac{1}{a} + \frac{1}{b}}{b - a}} \]
7. Step-by-step derivation

   1. *-commutative 71.5%

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

   2. +-commutative 71.5%

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

   3. associate-/l* 71.5%

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

   4. +-commutative 71.5%

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

8. Simplified 71.5%

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

9. Taylor expanded in b around inf 99.6%

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

10. Taylor expanded in a around inf 64.8%

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

    1. associate-*l/ 64.9%

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

    2. *-un-lft-identity 64.9%

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

12. Applied egg-rr 64.9%

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

13. Final simplification 64.9%

    \[\leadsto \frac{\pi \cdot \frac{0.5}{a}}{a \cdot b} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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))))