NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.6%

Time: 10.8s

Alternatives: 9

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: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. un-div-inv 78.3%

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

   2. difference-of-squares 86.5%

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

   3. associate-/r* 87.7%

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

   4. div-inv 87.7%

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

   5. metadata-eval 87.7%

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

4. Applied egg-rr 87.7%

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

   1. frac-sub 87.7%

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

   2. frac-times 99.5%

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

   3. associate-/l* 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. *-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. associate-/l* 99.4%

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

   3. associate-*r/ 99.4%

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

   4. *-commutative 99.4%

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

   5. +-commutative 99.4%

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

   6. *-commutative 99.4%

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

8. Simplified 99.4%

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

   1. clear-num 99.4%

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

   2. +-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. clear-num 99.4%

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

   5. frac-times 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. *-commutative 99.4%

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

   9. times-frac 99.4%

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

   10. metadata-eval 99.4%

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

   11. +-commutative 99.4%

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

   12. *-commutative 99.4%

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

   13. *-commutative 99.4%

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

   14. *-un-lft-identity 99.4%

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

   15. times-frac 99.4%

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

   16. *-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

    1. *-rgt-identity 99.4%

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

    2. /-rgt-identity 99.4%

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

    3. associate-/r* 99.7%

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

    4. associate-/r* 99.7%

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

    5. metadata-eval 99.7%

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

12. Simplified 99.7%

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

Alternative 2: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a + b}}{a} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+150)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ (/ 0.5 (+ a b)) a) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+150) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((0.5 / (a + b)) / a) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e+150) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((0.5 / (a + b)) / a) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e+150:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((0.5 / (a + b)) / a) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+150)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(0.5 / Float64(a + b)) / a) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+150)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((0.5 / (a + b)) / a) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+150], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999996e150

   1. Initial program 46.2%

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

      1. un-div-inv 46.2%

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

      2. difference-of-squares 72.7%

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

      3. associate-/r* 74.1%

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

      4. div-inv 74.1%

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

      5. metadata-eval 74.1%

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

   4. Applied egg-rr 74.1%

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

      1. frac-sub 74.1%

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

      2. frac-times 99.8%

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

      3. associate-/l* 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-rgt-identity 99.8%

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

      2. associate-/l* 99.7%

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

      3. associate-*r/ 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. *-un-lft-identity 99.9%

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

      8. *-commutative 99.9%

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

      9. times-frac 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. *-commutative 99.9%

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

      14. *-un-lft-identity 99.9%

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

      15. times-frac 99.8%

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

      16. *-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. *-rgt-identity 99.8%

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

       2. /-rgt-identity 99.8%

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

       3. associate-/r* 99.7%

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

       4. associate-/r* 99.7%

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

       5. metadata-eval 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in a around inf 99.8%

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

   if -1.99999999999999996e150 < a

   1. Initial program 83.2%

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

      1. un-div-inv 83.2%

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

      2. difference-of-squares 88.7%

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

      3. associate-/r* 89.8%

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

      4. div-inv 89.8%

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

      5. metadata-eval 89.8%

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

   4. Applied egg-rr 89.8%

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

      1. frac-sub 89.8%

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

      2. frac-times 99.4%

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

      3. associate-/l* 99.4%

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

      4. *-un-lft-identity 99.4%

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

      5. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-rgt-identity 99.4%

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

      2. associate-/l* 99.3%

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

      3. associate-*r/ 99.4%

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

      4. *-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

   8. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. +-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. clear-num 99.4%

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

      5. frac-times 99.3%

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

      6. metadata-eval 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. *-commutative 99.3%

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

      9. times-frac 99.3%

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

      10. metadata-eval 99.3%

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

      11. +-commutative 99.3%

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

      12. *-commutative 99.3%

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

      13. *-commutative 99.3%

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

      14. *-un-lft-identity 99.3%

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

      15. times-frac 99.4%

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

      16. *-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. *-rgt-identity 99.4%

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

       2. /-rgt-identity 99.4%

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

       3. associate-/r* 99.7%

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

       4. associate-/r* 99.7%

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

       5. metadata-eval 99.7%

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

   12. Simplified 99.7%

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

       1. associate-/r/ 99.6%

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

       2. times-frac 97.4%

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

   14. Applied egg-rr 97.4%

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

Alternative 3: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.55 \cdot 10^{+103}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a + b}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.55e+103)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ 0.5 b) (/ (/ PI (+ a b)) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.55e+103) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / (a + b)) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.55e+103) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 / b) * ((Math.PI / (a + b)) / a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.55e+103:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 / b) * ((math.pi / (a + b)) / a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.55e+103)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / Float64(a + b)) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.55e+103)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 / b) * ((pi / (a + b)) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.55e+103], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5500000000000001e103

   1. Initial program 50.5%

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

      1. un-div-inv 50.5%

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

      2. difference-of-squares 74.8%

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

      3. associate-/r* 76.1%

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

      4. div-inv 76.1%

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

      5. metadata-eval 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. frac-sub 76.1%

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

      2. frac-times 99.7%

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

      3. associate-/l* 99.7%

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

      4. *-un-lft-identity 99.7%

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

      5. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. associate-/l* 99.7%

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

      3. associate-*r/ 99.7%

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

      4. *-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. frac-times 99.8%

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

      6. metadata-eval 99.8%

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

      7. *-un-lft-identity 99.8%

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

      8. *-commutative 99.8%

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

      9. times-frac 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

      12. *-commutative 99.8%

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

      13. *-commutative 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. times-frac 99.7%

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

      16. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-rgt-identity 99.7%

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

       2. /-rgt-identity 99.7%

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

       3. associate-/r* 99.7%

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

       4. associate-/r* 99.7%

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

       5. metadata-eval 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in a around inf 99.7%

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

   if -2.5500000000000001e103 < a

   1. Initial program 83.0%

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

      1. un-div-inv 83.0%

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

      2. difference-of-squares 88.5%

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

      3. associate-/r* 89.7%

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

      4. div-inv 89.7%

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

      5. metadata-eval 89.7%

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

   4. Applied egg-rr 89.7%

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

      1. frac-sub 89.7%

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

      2. frac-times 99.4%

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

      3. associate-/l* 99.4%

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

      4. *-un-lft-identity 99.4%

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

      5. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-rgt-identity 99.4%

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

      2. associate-/l* 99.3%

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

      3. associate-*r/ 99.4%

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

      4. *-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

   8. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. +-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. clear-num 99.4%

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

      5. frac-times 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. *-commutative 99.4%

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

      9. times-frac 99.4%

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

      10. metadata-eval 99.4%

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

      11. +-commutative 99.4%

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

      12. *-commutative 99.4%

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

      13. *-commutative 99.4%

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

      14. *-un-lft-identity 99.4%

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

      15. times-frac 99.4%

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

      16. *-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. *-rgt-identity 99.4%

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

       2. /-rgt-identity 99.4%

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

       3. associate-/r* 99.7%

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

       4. associate-/r* 99.7%

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

       5. metadata-eval 99.7%

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

   12. Simplified 99.7%

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

       1. div-inv 99.7%

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

       2. *-commutative 99.7%

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

       3. times-frac 97.4%

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

       4. clear-num 97.4%

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

   14. Applied egg-rr 97.4%

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

Alternative 4: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e-96)
   (/ (* 0.5 PI) (* a (* a b)))
   (/ (* 0.5 (/ PI (* a b))) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-96) {
		tmp = (0.5 * ((double) M_PI)) / (a * (a * b));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-96) {
		tmp = (0.5 * Math.PI) / (a * (a * b));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4e-96:
		tmp = (0.5 * math.pi) / (a * (a * b))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e-96)
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4e-96)
		tmp = (0.5 * pi) / (a * (a * b));
	else
		tmp = (0.5 * (pi / (a * b))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e-96], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000019e-96

   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. Add Preprocessing
   3. Step-by-step derivation

      1. un-div-inv 75.1%

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

      2. difference-of-squares 86.6%

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

      3. associate-/r* 87.2%

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

      4. div-inv 87.2%

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

      5. metadata-eval 87.2%

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

   4. Applied egg-rr 87.2%

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

      1. frac-sub 87.2%

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

      2. frac-times 99.3%

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

      3. associate-/l* 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. *-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-rgt-identity 99.1%

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

      2. associate-/l* 99.2%

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

      3. associate-*r/ 99.3%

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

      4. *-commutative 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

   8. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.6%

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

      3. pow1 99.6%

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

      4. pow1 99.6%

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

      5. pow-div 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. frac-times 99.4%

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

      11. *-un-lft-identity 99.4%

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

      12. *-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in a around inf 84.7%

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

   if -2.40000000000000019e-96 < a

   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. Add Preprocessing
   3. Step-by-step derivation

      1. un-div-inv 79.7%

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

      2. difference-of-squares 86.5%

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

      3. associate-/r* 87.9%

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

      4. div-inv 87.9%

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

      5. metadata-eval 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. *-commutative 87.9%

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

      2. frac-sub 87.9%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. associate-/l* 99.5%

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

      6. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r* 98.9%

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

      2. *-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*r* 95.8%

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

      5. times-frac 87.0%

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

      6. *-rgt-identity 87.0%

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

      7. +-commutative 87.0%

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

   8. Applied egg-rr 87.0%

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

      1. associate-*r/ 95.9%

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

      2. times-frac 96.9%

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

      3. *-inverses 96.9%

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

      4. *-lft-identity 96.9%

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

   10. Simplified 96.9%

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

   11. Taylor expanded in a around 0 76.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.5e-96)
   (/ (* 0.5 PI) (* a (* a b)))
   (/ (* 0.5 PI) (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.5e-96) {
		tmp = (0.5 * ((double) M_PI)) / (a * (a * b));
	} 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 <= -2.5e-96) {
		tmp = (0.5 * Math.PI) / (a * (a * b));
	} else {
		tmp = (0.5 * Math.PI) / (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.5e-96:
		tmp = (0.5 * math.pi) / (a * (a * b))
	else:
		tmp = (0.5 * math.pi) / (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.5e-96)
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.5e-96)
		tmp = (0.5 * pi) / (a * (a * b));
	else
		tmp = (0.5 * pi) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.5e-96], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.49999999999999997e-96

   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. Add Preprocessing
   3. Step-by-step derivation

      1. un-div-inv 75.1%

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

      2. difference-of-squares 86.6%

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

      3. associate-/r* 87.2%

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

      4. div-inv 87.2%

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

      5. metadata-eval 87.2%

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

   4. Applied egg-rr 87.2%

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

      1. frac-sub 87.2%

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

      2. frac-times 99.3%

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

      3. associate-/l* 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. *-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-rgt-identity 99.1%

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

      2. associate-/l* 99.2%

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

      3. associate-*r/ 99.3%

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

      4. *-commutative 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

   8. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.6%

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

      3. pow1 99.6%

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

      4. pow1 99.6%

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

      5. pow-div 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. frac-times 99.4%

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

      11. *-un-lft-identity 99.4%

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

      12. *-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in a around inf 84.7%

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

   if -2.49999999999999997e-96 < a

   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. Add Preprocessing
   3. Step-by-step derivation

      1. un-div-inv 79.7%

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

      2. difference-of-squares 86.5%

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

      3. associate-/r* 87.9%

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

      4. div-inv 87.9%

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

      5. metadata-eval 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. frac-sub 87.9%

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

      2. frac-times 99.5%

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

      3. associate-/l* 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. *-rgt-identity 99.5%

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

      2. associate-/l* 99.5%

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

      3. associate-*r/ 99.5%

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

      4. *-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. *-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.7%

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

      3. pow1 99.7%

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

      4. pow1 99.7%

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

      5. pow-div 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. *-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. frac-times 99.6%

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

      11. *-un-lft-identity 99.6%

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

      12. *-commutative 99.6%

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

      13. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around 0 76.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. un-div-inv 78.3%

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

   2. difference-of-squares 86.5%

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

   3. associate-/r* 87.7%

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

   4. div-inv 87.7%

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

   5. metadata-eval 87.7%

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

4. Applied egg-rr 87.7%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.7%

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

    1. un-div-inv 99.7%

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

    2. div-inv 99.7%

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

    3. *-commutative 99.7%

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

    4. associate-*r* 99.7%

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

    5. div-inv 99.7%

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

    6. frac-times 93.1%

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

    7. clear-num 93.0%

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

    8. frac-times 99.6%

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

    9. *-un-lft-identity 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 7: 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 78.3%

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

   1. un-div-inv 78.3%

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

   2. difference-of-squares 86.5%

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

   3. associate-/r* 87.7%

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

   4. div-inv 87.7%

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

   5. metadata-eval 87.7%

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

4. Applied egg-rr 87.7%

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

   1. frac-sub 87.7%

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

   2. frac-times 99.5%

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

   3. associate-/l* 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. *-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. associate-/l* 99.4%

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

   3. associate-*r/ 99.4%

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

   4. *-commutative 99.4%

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

   5. +-commutative 99.4%

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

   6. *-commutative 99.4%

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

8. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. clear-num 99.4%

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

   3. +-commutative 99.4%

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

   4. frac-times 99.5%

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

   5. *-un-lft-identity 99.5%

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

   6. *-commutative 99.5%

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

   7. *-commutative 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. times-frac 99.5%

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

   10. *-commutative 99.5%

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

   11. pow1 99.5%

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

   12. pow1 99.5%

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

   13. pow-div 99.5%

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

   14. metadata-eval 99.5%

       \[\leadsto \frac{0.5 \cdot \pi}{\left(\frac{a \cdot b}{1} \cdot {\left(b - a\right)}^{\color{blue}{0}}\right) \cdot \left(b + a\right)} \]

   15. metadata-eval 99.5%

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

   16. +-commutative 99.5%

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

10. Applied egg-rr 99.5%

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

    1. *-rgt-identity 99.5%

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

    2. /-rgt-identity 99.5%

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

    3. *-commutative 99.5%

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

    4. associate-/r* 99.7%

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

    5. associate-*l/ 99.7%

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

    6. *-commutative 99.7%

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

12. Simplified 99.7%

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

Alternative 8: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. un-div-inv 78.3%

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

   2. difference-of-squares 86.5%

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

   3. associate-/r* 87.7%

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

   4. div-inv 87.7%

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

   5. metadata-eval 87.7%

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

4. Applied egg-rr 87.7%

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

   1. frac-sub 87.7%

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

   2. frac-times 99.5%

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

   3. associate-/l* 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. *-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. associate-/l* 99.4%

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

   3. associate-*r/ 99.4%

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

   4. *-commutative 99.4%

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

   5. +-commutative 99.4%

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

   6. *-commutative 99.4%

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

8. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-/r* 99.7%

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

   3. pow1 99.7%

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

   4. pow1 99.7%

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

   5. pow-div 99.7%

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

   6. metadata-eval 99.7%

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

   7. metadata-eval 99.7%

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

   8. *-commutative 99.7%

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

   9. +-commutative 99.7%

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

   10. frac-times 99.5%

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

   11. *-un-lft-identity 99.5%

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

   12. *-commutative 99.5%

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

   13. +-commutative 99.5%

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 9: 62.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. un-div-inv 78.3%

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

   2. difference-of-squares 86.5%

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

   3. associate-/r* 87.7%

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

   4. div-inv 87.7%

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

   5. metadata-eval 87.7%

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

4. Applied egg-rr 87.7%

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

   1. frac-sub 87.7%

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

   2. frac-times 99.5%

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

   3. associate-/l* 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. *-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. associate-/l* 99.4%

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

   3. associate-*r/ 99.4%

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

   4. *-commutative 99.4%

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

   5. +-commutative 99.4%

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

   6. *-commutative 99.4%

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

8. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-/r* 99.7%

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

   3. pow1 99.7%

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

   4. pow1 99.7%

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

   5. pow-div 99.7%

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

   6. metadata-eval 99.7%

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

   7. metadata-eval 99.7%

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

   8. *-commutative 99.7%

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

   9. +-commutative 99.7%

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

   10. frac-times 99.5%

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

   11. *-un-lft-identity 99.5%

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

   12. *-commutative 99.5%

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

   13. +-commutative 99.5%

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

10. Applied egg-rr 99.5%

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

11. Taylor expanded in a around inf 62.1%

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

12. Final simplification 62.1%

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

Reproduce

herbie shell --seed 2024132 
(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))))