NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.7%

Time: 14.2s

Alternatives: 9

Speedup: 1.1×

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: 78.8% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. add-sqr-sqrt 51.7%

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

   2. sqrt-unprod 47.8%

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

   3. pow1 47.8%

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

   4. pow1 47.8%

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

   5. pow-sqr 47.8%

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

3. Applied egg-rr 53.5%

   \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ 53.5%

      \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

   2. associate-*l* 53.5%

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

   3. +-commutative 53.5%

      \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

   4. *-commutative 53.5%

      \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
6. Step-by-step derivation

   1. sqrt-pow1 99.6%

      \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

   3. metadata-eval 99.6%

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

   4. pow1 99.6%

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

   5. associate-*r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. div-inv 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-*l/ 99.6%

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

   10. frac-times 99.6%

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

   11. *-commutative 99.6%

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

   12. un-div-inv 99.7%

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

   13. +-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 96.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(\left(b + a\right) \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e+106)
   (/ (/ (* PI (/ 0.5 a)) b) a)
   (/ PI (* b (* a (* (+ b a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e+106) {
		tmp = ((((double) M_PI) * (0.5 / a)) / b) / a;
	} else {
		tmp = ((double) M_PI) / (b * (a * ((b + a) * 2.0)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9e+106) {
		tmp = ((Math.PI * (0.5 / a)) / b) / a;
	} else {
		tmp = Math.PI / (b * (a * ((b + a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e+106:
		tmp = ((math.pi * (0.5 / a)) / b) / a
	else:
		tmp = math.pi / (b * (a * ((b + a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e+106)
		tmp = Float64(Float64(Float64(pi * Float64(0.5 / a)) / b) / a);
	else
		tmp = Float64(pi / Float64(b * Float64(a * Float64(Float64(b + a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e+106)
		tmp = ((pi * (0.5 / a)) / b) / a;
	else
		tmp = pi / (b * (a * ((b + a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e+106], N[(N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], N[(Pi / N[(b * N[(a * N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.9999999999999994e106

   1. Initial program 59.8%

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

      1. add-sqr-sqrt 57.1%

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

      2. sqrt-unprod 55.3%

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

      3. pow1 55.3%

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

      4. pow1 55.3%

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

      5. pow-sqr 55.3%

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

   3. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

      2. associate-*l* 73.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

      3. +-commutative 73.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

      4. *-commutative 73.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 99.6%

         \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

      3. metadata-eval 99.6%

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

      4. pow1 99.6%

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

      5. associate-/r* 99.7%

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

      6. un-div-inv 99.7%

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

      7. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in b around 0 99.7%

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

   if -8.9999999999999994e106 < a

   1. Initial program 78.7%

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

      1. associate-*l* 78.6%

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

      2. associate-*l/ 78.6%

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

      3. frac-sub 78.6%

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

      4. *-commutative 78.6%

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

      5. associate-*r/ 78.5%

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

      6. *-commutative 78.5%

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

      7. *-un-lft-identity 78.5%

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

      8. *-un-lft-identity 78.5%

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

      9. associate-/r/ 78.6%

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

      10. flip-+ 99.5%

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

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{\frac{1}{b + a}}{b \cdot a}}{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 98.9%

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

      2. associate-/l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. metadata-eval 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*l* 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-*l* 94.8%

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

      9. +-commutative 94.8%

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

   5. Simplified 94.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot \left(\left(b + a\right) \cdot 2\right)\right)}\\ \end{array} \]

Alternative 3: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.5e-81) (/ (* 0.5 (/ (/ PI b) a)) a) (* PI (/ (/ 0.5 (* b a)) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-81) {
		tmp = (0.5 * ((((double) M_PI) / b) / a)) / a;
	} else {
		tmp = ((double) M_PI) * ((0.5 / (b * a)) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-81) {
		tmp = (0.5 * ((Math.PI / b) / a)) / a;
	} else {
		tmp = Math.PI * ((0.5 / (b * a)) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.5e-81:
		tmp = (0.5 * ((math.pi / b) / a)) / a
	else:
		tmp = math.pi * ((0.5 / (b * a)) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.5e-81)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / b) / a)) / a);
	else
		tmp = Float64(pi * Float64(Float64(0.5 / Float64(b * a)) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.5e-81)
		tmp = (0.5 * ((pi / b) / a)) / a;
	else
		tmp = pi * ((0.5 / (b * a)) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.5e-81], N[(N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(Pi * N[(N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.49999999999999917e-81

   1. Initial program 74.9%

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

      1. add-sqr-sqrt 46.9%

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

      2. sqrt-unprod 44.9%

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

      3. pow1 44.9%

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

      4. pow1 44.9%

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

      5. pow-sqr 44.9%

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

   3. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 50.4%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

      2. associate-*l* 50.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

      3. +-commutative 50.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

      4. *-commutative 50.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

   5. Simplified 50.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 99.6%

         \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

      3. metadata-eval 99.6%

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

      4. pow1 99.6%

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

      5. associate-/r* 94.5%

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

      6. un-div-inv 94.5%

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

      7. +-commutative 94.5%

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

   7. Applied egg-rr 94.5%

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

   8. Taylor expanded in b around 0 73.7%

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

      1. associate-/l/ 73.7%

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

   10. Simplified 73.7%

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

   if 9.49999999999999917e-81 < b

   1. Initial program 78.0%

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

      1. associate-*l* 77.9%

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

      2. associate-*l/ 77.9%

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

      3. frac-sub 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-*r/ 77.8%

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

      6. *-commutative 77.8%

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

      7. *-un-lft-identity 77.8%

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

      8. *-un-lft-identity 77.8%

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

      9. associate-/r/ 77.9%

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

      10. flip-+ 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{\frac{1}{b + a}}{b \cdot a}}{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 97.0%

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

      2. associate-/l/ 96.9%

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

      3. associate-/r/ 97.0%

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

      4. metadata-eval 97.0%

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

      5. *-commutative 97.0%

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

      6. associate-*l* 97.0%

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

      7. *-commutative 97.0%

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

      8. associate-*l* 95.9%

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

      9. +-commutative 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in a around 0 84.4%

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

      1. associate-*r* 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

   8. Simplified 84.4%

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

      1. div-inv 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

      4. associate-/r* 84.7%

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

      5. associate-*r* 84.7%

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

      6. *-commutative 84.7%

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

      7. associate-/r* 84.7%

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

      8. metadata-eval 84.7%

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

   10. Applied egg-rr 84.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{b}\\ \end{array} \]

Alternative 4: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{b}}{a}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (/ PI b) a))))
   (if (<= a -1.8e-66) (/ t_0 a) (/ t_0 b))))

C

double code(double a, double b) {
	double t_0 = 0.5 * ((((double) M_PI) / b) / a);
	double tmp;
	if (a <= -1.8e-66) {
		tmp = t_0 / a;
	} else {
		tmp = t_0 / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 * ((Math.PI / b) / a);
	double tmp;
	if (a <= -1.8e-66) {
		tmp = t_0 / a;
	} else {
		tmp = t_0 / b;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 * ((math.pi / b) / a)
	tmp = 0
	if a <= -1.8e-66:
		tmp = t_0 / a
	else:
		tmp = t_0 / b
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 * Float64(Float64(pi / b) / a))
	tmp = 0.0
	if (a <= -1.8e-66)
		tmp = Float64(t_0 / a);
	else
		tmp = Float64(t_0 / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 * ((pi / b) / a);
	tmp = 0.0;
	if (a <= -1.8e-66)
		tmp = t_0 / a;
	else
		tmp = t_0 / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.8e-66], N[(t$95$0 / a), $MachinePrecision], N[(t$95$0 / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.80000000000000006e-66

   1. Initial program 77.2%

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

      1. add-sqr-sqrt 51.0%

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

      2. sqrt-unprod 46.8%

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

      3. pow1 46.8%

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

      4. pow1 46.8%

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

      5. pow-sqr 46.8%

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

   3. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 56.3%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

      2. associate-*l* 56.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

      3. +-commutative 56.3%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

      4. *-commutative 56.3%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 99.6%

         \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

      3. metadata-eval 99.6%

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

      4. pow1 99.6%

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

      5. associate-/r* 99.1%

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

      6. un-div-inv 99.1%

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

      7. +-commutative 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in b around 0 84.1%

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

      1. associate-/l/ 84.0%

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

   10. Simplified 84.0%

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

   if -1.80000000000000006e-66 < a

   1. Initial program 75.4%

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

      1. associate-*l* 75.3%

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

      2. associate-*l/ 75.3%

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

      3. frac-sub 75.3%

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

      4. *-commutative 75.3%

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

      5. associate-*r/ 75.2%

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

      6. *-commutative 75.2%

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

      7. *-un-lft-identity 75.2%

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

      8. *-un-lft-identity 75.2%

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

      9. associate-/r/ 75.2%

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

      10. flip-+ 99.5%

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

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{\frac{1}{b + a}}{b \cdot a}}{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 98.8%

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

      2. associate-/l/ 98.9%

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

      3. associate-/r/ 99.0%

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

      4. metadata-eval 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*l* 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-*l* 93.9%

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

      9. +-commutative 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in a around 0 67.7%

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

      1. associate-*r* 67.7%

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

      2. *-commutative 67.7%

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

      3. *-commutative 67.7%

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

   8. Simplified 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. times-frac 67.9%

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

      3. associate-*r* 67.9%

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

      4. *-commutative 67.9%

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

      5. associate-*l* 67.9%

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

   10. Applied egg-rr 67.9%

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

       1. associate-*l/ 67.9%

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

       2. associate-*r* 67.9%

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

       3. *-commutative 67.9%

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

       4. associate-/l/ 67.9%

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

       5. metadata-eval 67.9%

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

       6. associate-/l* 67.9%

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

       7. /-rgt-identity 67.9%

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

       8. *-commutative 67.9%

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

       9. associate-*r/ 67.9%

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

       10. *-commutative 67.9%

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

       11. *-lft-identity 67.9%

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

       12. associate-/l/ 67.9%

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

   12. Simplified 67.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{b}\\ \end{array} \]

Alternative 5: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{b \cdot a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.2e-64)
   (/ (/ 0.5 a) (/ (* b a) PI))
   (/ (* 0.5 (/ (/ PI b) a)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.2e-64) {
		tmp = (0.5 / a) / ((b * a) / ((double) M_PI));
	} else {
		tmp = (0.5 * ((((double) M_PI) / b) / a)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.2e-64) {
		tmp = (0.5 / a) / ((b * a) / Math.PI);
	} else {
		tmp = (0.5 * ((Math.PI / b) / a)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.2e-64:
		tmp = (0.5 / a) / ((b * a) / math.pi)
	else:
		tmp = (0.5 * ((math.pi / b) / a)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.2e-64)
		tmp = Float64(Float64(0.5 / a) / Float64(Float64(b * a) / pi));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / b) / a)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.2e-64)
		tmp = (0.5 / a) / ((b * a) / pi);
	else
		tmp = (0.5 * ((pi / b) / a)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.2e-64], N[(N[(0.5 / a), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.2000000000000001e-64

   1. Initial program 78.2%

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

      1. add-sqr-sqrt 51.6%

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

      2. sqrt-unprod 47.3%

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

      3. pow1 47.3%

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

      4. pow1 47.3%

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

      5. pow-sqr 47.3%

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

   3. Applied egg-rr 57.0%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 57.0%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

      2. associate-*l* 57.0%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

      3. +-commutative 57.0%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

      4. *-commutative 57.0%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 99.6%

         \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

      3. metadata-eval 99.6%

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

      4. pow1 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/l* 99.7%

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

      7. un-div-inv 99.7%

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

      8. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in b around 0 85.2%

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

   if -8.2000000000000001e-64 < a

   1. Initial program 75.0%

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

      1. associate-*l* 74.9%

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

      2. associate-*l/ 74.9%

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

      3. frac-sub 74.9%

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

      4. *-commutative 74.9%

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

      5. associate-*r/ 74.8%

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

      6. *-commutative 74.8%

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

      7. *-un-lft-identity 74.8%

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

      8. *-un-lft-identity 74.8%

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

      9. associate-/r/ 74.9%

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

      10. flip-+ 99.5%

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

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{\frac{1}{b + a}}{b \cdot a}}{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 98.8%

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

      2. associate-/l/ 98.9%

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

      3. associate-/r/ 99.0%

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

      4. metadata-eval 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*l* 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-*l* 94.0%

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

      9. +-commutative 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in a around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. *-commutative 67.9%

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

      3. *-commutative 67.9%

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

   8. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. times-frac 68.0%

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

      3. associate-*r* 68.0%

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

      4. *-commutative 68.0%

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

      5. associate-*l* 68.0%

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

   10. Applied egg-rr 68.0%

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

       1. associate-*l/ 68.1%

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

       2. associate-*r* 68.1%

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

       3. *-commutative 68.1%

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

       4. associate-/l/ 68.1%

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

       5. metadata-eval 68.1%

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

       6. associate-/l* 68.1%

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

       7. /-rgt-identity 68.1%

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

       8. *-commutative 68.1%

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

       9. associate-*r/ 68.1%

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

       10. *-commutative 68.1%

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

       11. *-lft-identity 68.1%

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

       12. associate-/l/ 68.1%

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

   12. Simplified 68.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{b \cdot a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{b}\\ \end{array} \]

Alternative 6: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.7e-81) (/ (* 0.5 (/ (/ PI b) a)) a) (/ (/ (* PI 0.5) b) (* b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.7e-81) {
		tmp = (0.5 * ((((double) M_PI) / b) / a)) / a;
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) / (b * a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7.7e-81) {
		tmp = (0.5 * ((Math.PI / b) / a)) / a;
	} else {
		tmp = ((Math.PI * 0.5) / b) / (b * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.7e-81:
		tmp = (0.5 * ((math.pi / b) / a)) / a
	else:
		tmp = ((math.pi * 0.5) / b) / (b * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.7e-81)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / b) / a)) / a);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) / Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7.7e-81)
		tmp = (0.5 * ((pi / b) / a)) / a;
	else
		tmp = ((pi * 0.5) / b) / (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.7e-81], N[(N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.7000000000000002e-81

   1. Initial program 74.9%

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

      1. add-sqr-sqrt 46.9%

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

      2. sqrt-unprod 44.9%

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

      3. pow1 44.9%

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

      4. pow1 44.9%

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

      5. pow-sqr 44.9%

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

   3. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 50.4%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

      2. associate-*l* 50.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

      3. +-commutative 50.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

      4. *-commutative 50.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

   5. Simplified 50.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 99.6%

         \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

      3. metadata-eval 99.6%

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

      4. pow1 99.6%

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

      5. associate-/r* 94.5%

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

      6. un-div-inv 94.5%

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

      7. +-commutative 94.5%

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

   7. Applied egg-rr 94.5%

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

   8. Taylor expanded in b around 0 73.7%

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

      1. associate-/l/ 73.7%

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

   10. Simplified 73.7%

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

   if 7.7000000000000002e-81 < b

   1. Initial program 78.0%

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

      1. associate-*l* 77.9%

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

      2. associate-*l/ 77.9%

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

      3. frac-sub 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-*r/ 77.8%

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

      6. *-commutative 77.8%

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

      7. *-un-lft-identity 77.8%

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

      8. *-un-lft-identity 77.8%

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

      9. associate-/r/ 77.9%

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

      10. flip-+ 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{\frac{1}{b + a}}{b \cdot a}}{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 97.0%

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

      2. associate-/l/ 96.9%

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

      3. associate-/r/ 97.0%

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

      4. metadata-eval 97.0%

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

      5. *-commutative 97.0%

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

      6. associate-*l* 97.0%

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

      7. *-commutative 97.0%

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

      8. associate-*l* 95.9%

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

      9. +-commutative 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in a around 0 84.4%

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

      1. associate-*r* 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

   8. Simplified 84.4%

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

      1. div-inv 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

      4. associate-/r* 84.7%

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

      5. associate-*r* 84.7%

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

      6. *-commutative 84.7%

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

      7. associate-/r* 84.7%

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

      8. metadata-eval 84.7%

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

   10. Applied egg-rr 84.7%

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

       1. associate-*l/ 84.7%

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

       2. associate-*l/ 84.7%

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

       3. associate-*r/ 84.7%

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

       4. associate-/r* 84.8%

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

       5. associate-*r/ 84.8%

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

       6. associate-/l/ 84.8%

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

       7. associate-*r/ 84.8%

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

   12. Applied egg-rr 84.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{b \cdot a}\\ \end{array} \]

Alternative 7: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e-82) (/ (/ (* PI (/ 0.5 a)) b) a) (/ (/ (* PI 0.5) b) (* b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-82) {
		tmp = ((((double) M_PI) * (0.5 / a)) / b) / a;
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) / (b * a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-82) {
		tmp = ((Math.PI * (0.5 / a)) / b) / a;
	} else {
		tmp = ((Math.PI * 0.5) / b) / (b * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.9e-82:
		tmp = ((math.pi * (0.5 / a)) / b) / a
	else:
		tmp = ((math.pi * 0.5) / b) / (b * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.9e-82)
		tmp = Float64(Float64(Float64(pi * Float64(0.5 / a)) / b) / a);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) / Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.9e-82)
		tmp = ((pi * (0.5 / a)) / b) / a;
	else
		tmp = ((pi * 0.5) / b) / (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.9e-82], N[(N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.89999999999999977e-82

   1. Initial program 74.9%

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

      1. add-sqr-sqrt 46.9%

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

      2. sqrt-unprod 44.9%

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

      3. pow1 44.9%

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

      4. pow1 44.9%

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

      5. pow-sqr 44.9%

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

   3. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 50.4%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

      2. associate-*l* 50.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

      3. +-commutative 50.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

      4. *-commutative 50.4%

         \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

   5. Simplified 50.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-pow1 99.6%

         \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

      3. metadata-eval 99.6%

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

      4. pow1 99.6%

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

      5. associate-/r* 94.5%

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

      6. un-div-inv 94.5%

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

      7. +-commutative 94.5%

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

   7. Applied egg-rr 94.5%

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

   8. Taylor expanded in b around 0 73.6%

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

   if 2.89999999999999977e-82 < b

   1. Initial program 78.0%

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

      1. associate-*l* 77.9%

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

      2. associate-*l/ 77.9%

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

      3. frac-sub 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-*r/ 77.8%

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

      6. *-commutative 77.8%

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

      7. *-un-lft-identity 77.8%

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

      8. *-un-lft-identity 77.8%

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

      9. associate-/r/ 77.9%

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

      10. flip-+ 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{\frac{1}{b + a}}{b \cdot a}}{2}} \]
   4. Step-by-step derivation

      1. associate-/l* 97.0%

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

      2. associate-/l/ 96.9%

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

      3. associate-/r/ 97.0%

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

      4. metadata-eval 97.0%

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

      5. *-commutative 97.0%

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

      6. associate-*l* 97.0%

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

      7. *-commutative 97.0%

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

      8. associate-*l* 95.9%

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

      9. +-commutative 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in a around 0 84.4%

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

      1. associate-*r* 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

   8. Simplified 84.4%

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

      1. div-inv 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

      4. associate-/r* 84.7%

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

      5. associate-*r* 84.7%

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

      6. *-commutative 84.7%

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

      7. associate-/r* 84.7%

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

      8. metadata-eval 84.7%

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

   10. Applied egg-rr 84.7%

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

       1. associate-*l/ 84.7%

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

       2. associate-*l/ 84.7%

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

       3. associate-*r/ 84.7%

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

       4. associate-/r* 84.8%

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

       5. associate-*r/ 84.8%

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

       6. associate-/l/ 84.8%

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

       7. associate-*r/ 84.8%

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

   12. Applied egg-rr 84.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{b \cdot a}\\ \end{array} \]

Alternative 8: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. add-sqr-sqrt 51.7%

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

   2. sqrt-unprod 47.8%

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

   3. pow1 47.8%

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

   4. pow1 47.8%

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

   5. pow-sqr 47.8%

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

3. Applied egg-rr 53.5%

   \[\leadsto \color{blue}{\sqrt{{\left(\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{b + a}}{b \cdot a}\right)}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ 53.5%

      \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b + a}}{b \cdot a}\right)}}^{2}} \]

   2. associate-*l* 53.5%

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b + a}\right)}}{b \cdot a}\right)}^{2}} \]

   3. +-commutative 53.5%

      \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{\color{blue}{a + b}}\right)}{b \cdot a}\right)}^{2}} \]

   4. *-commutative 53.5%

      \[\leadsto \sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{a \cdot b}}\right)}^{2}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{\sqrt{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{2}}} \]
6. Step-by-step derivation

   1. sqrt-pow1 99.6%

      \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{a \cdot b}\right)}^{\left(\frac{2}{2}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto {\left(\frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{b \cdot a}}\right)}^{\left(\frac{2}{2}\right)} \]

   3. metadata-eval 99.6%

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

   4. pow1 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-/l* 99.6%

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

   7. un-div-inv 99.6%

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

   8. +-commutative 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 9: 62.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-*l* 75.8%

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

   2. associate-*l/ 75.8%

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

   3. frac-sub 75.8%

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

   4. *-commutative 75.8%

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

   5. associate-*r/ 75.7%

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

   6. *-commutative 75.7%

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

   7. *-un-lft-identity 75.7%

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

   8. *-un-lft-identity 75.7%

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

   9. associate-/r/ 75.8%

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

   10. flip-+ 99.6%

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

3. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\pi \cdot \frac{\frac{1}{b + a}}{b \cdot a}}{2}} \]
4. Step-by-step derivation

   1. associate-/l* 98.7%

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

   2. associate-/l/ 98.8%

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

   3. associate-/r/ 98.9%

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

   4. metadata-eval 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*l* 98.9%

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

   7. *-commutative 98.9%

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

   8. associate-*l* 91.2%

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

   9. +-commutative 91.2%

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

5. Simplified 91.2%

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

6. Taylor expanded in a around 0 60.5%

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

   1. associate-*r* 60.5%

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

   2. *-commutative 60.5%

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

   3. *-commutative 60.5%

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

8. Simplified 60.5%

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

   1. div-inv 60.4%

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

   2. *-commutative 60.4%

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

   3. *-commutative 60.4%

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

   4. associate-/r* 60.6%

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

   5. associate-*r* 60.6%

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

   6. *-commutative 60.6%

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

   7. associate-/r* 60.6%

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

   8. metadata-eval 60.6%

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

10. Applied egg-rr 60.6%

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

11. Final simplification 60.6%

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

Reproduce

herbie shell --seed 2023305 
(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))))