NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.6%

Time: 13.1s

Alternatives: 10

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. un-div-inv 80.6%

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

   2. difference-of-squares 89.5%

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

   3. associate-/r* 89.8%

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

   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. associate-*l/ 99.7%

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

   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* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. *-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. metadata-eval 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \]
10. Add Preprocessing

Alternative 2: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-198} \lor \neg \left(b \leq 7.6 \cdot 10^{-184}\right) \land b \leq 4.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b 3.5e-198) (and (not (<= b 7.6e-184)) (<= b 4.1e-76)))
   (* (/ (/ 1.0 a) b) (* 0.5 (/ PI a)))
   (* (/ 0.5 b) (/ PI (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= 3.5e-198) || (!(b <= 7.6e-184) && (b <= 4.1e-76))) {
		tmp = ((1.0 / a) / b) * (0.5 * (((double) M_PI) / a));
	} else {
		tmp = (0.5 / b) * (((double) M_PI) / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= 3.5e-198) || (!(b <= 7.6e-184) && (b <= 4.1e-76))) {
		tmp = ((1.0 / a) / b) * (0.5 * (Math.PI / a));
	} else {
		tmp = (0.5 / b) * (Math.PI / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= 3.5e-198) or (not (b <= 7.6e-184) and (b <= 4.1e-76)):
		tmp = ((1.0 / a) / b) * (0.5 * (math.pi / a))
	else:
		tmp = (0.5 / b) * (math.pi / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= 3.5e-198) || (!(b <= 7.6e-184) && (b <= 4.1e-76)))
		tmp = Float64(Float64(Float64(1.0 / a) / b) * Float64(0.5 * Float64(pi / a)));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(pi / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= 3.5e-198) || (~((b <= 7.6e-184)) && (b <= 4.1e-76)))
		tmp = ((1.0 / a) / b) * (0.5 * (pi / a));
	else
		tmp = (0.5 / b) * (pi / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, 3.5e-198], And[N[Not[LessEqual[b, 7.6e-184]], $MachinePrecision], LessEqual[b, 4.1e-76]]], N[(N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.50000000000000025e-198 or 7.60000000000000033e-184 < b < 4.0999999999999998e-76

   1. Initial program 80.3%

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

      1. un-div-inv 80.3%

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

      2. difference-of-squares 89.9%

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

      3. associate-/r* 90.3%

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

      4. div-inv 90.3%

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

      5. metadata-eval 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. associate-*l/ 99.7%

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

      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/ 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. sub-neg 99.7%

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

      6. distribute-neg-frac 99.7%

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

      7. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   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. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around inf 75.5%

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

   if 3.50000000000000025e-198 < b < 7.60000000000000033e-184 or 4.0999999999999998e-76 < b

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*r* 81.0%

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

      3. associate-*r/ 81.0%

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

      4. associate-*r* 81.0%

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

      5. *-rgt-identity 81.0%

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

      6. sub-neg 81.0%

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

      7. distribute-neg-frac 81.0%

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

      8. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. difference-of-squares 88.3%

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

      3. times-frac 98.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 81.9%

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

      6. frac-times 81.9%

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

      7. metadata-eval 81.9%

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

      8. metadata-eval 81.9%

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

      9. frac-times 81.9%

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

      10. sqrt-unprod 86.4%

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

      11. add-sqr-sqrt 86.4%

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

      12. div-inv 86.4%

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

      13. metadata-eval 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

      3. associate-/l* 87.6%

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

      4. associate-*l/ 87.7%

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

      5. +-commutative 87.7%

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

      6. +-commutative 87.7%

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

      7. *-commutative 87.7%

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

   8. Simplified 87.7%

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

   9. Taylor expanded in b around 0 87.6%

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

       1. associate-*l/ 87.7%

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

       2. *-un-lft-identity 87.7%

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

       3. associate-/l* 87.7%

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

   11. Applied egg-rr 87.7%

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

       1. *-commutative 87.7%

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

       2. times-frac 87.7%

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

   13. Simplified 87.7%

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

   14. Taylor expanded in b around inf 76.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-198} \lor \neg \left(b \leq 7.6 \cdot 10^{-184}\right) \land b \leq 4.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{a}}{b}\\ t_1 := t\_0 \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{if}\;b \leq 3.5 \cdot 10^{-198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \frac{\pi}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 a) b)) (t_1 (* t_0 (* 0.5 (/ PI a)))))
   (if (<= b 3.5e-198)
     t_1
     (if (<= b 7.6e-184)
       (* (/ 0.5 b) (/ PI (* a b)))
       (if (<= b 4.1e-76) t_1 (* t_0 (* 0.5 (/ PI b))))))))

C

double code(double a, double b) {
	double t_0 = (1.0 / a) / b;
	double t_1 = t_0 * (0.5 * (((double) M_PI) / a));
	double tmp;
	if (b <= 3.5e-198) {
		tmp = t_1;
	} else if (b <= 7.6e-184) {
		tmp = (0.5 / b) * (((double) M_PI) / (a * b));
	} else if (b <= 4.1e-76) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 * (((double) M_PI) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (1.0 / a) / b;
	double t_1 = t_0 * (0.5 * (Math.PI / a));
	double tmp;
	if (b <= 3.5e-198) {
		tmp = t_1;
	} else if (b <= 7.6e-184) {
		tmp = (0.5 / b) * (Math.PI / (a * b));
	} else if (b <= 4.1e-76) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 * (Math.PI / b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (1.0 / a) / b
	t_1 = t_0 * (0.5 * (math.pi / a))
	tmp = 0
	if b <= 3.5e-198:
		tmp = t_1
	elif b <= 7.6e-184:
		tmp = (0.5 / b) * (math.pi / (a * b))
	elif b <= 4.1e-76:
		tmp = t_1
	else:
		tmp = t_0 * (0.5 * (math.pi / b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(1.0 / a) / b)
	t_1 = Float64(t_0 * Float64(0.5 * Float64(pi / a)))
	tmp = 0.0
	if (b <= 3.5e-198)
		tmp = t_1;
	elseif (b <= 7.6e-184)
		tmp = Float64(Float64(0.5 / b) * Float64(pi / Float64(a * b)));
	elseif (b <= 4.1e-76)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(0.5 * Float64(pi / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (1.0 / a) / b;
	t_1 = t_0 * (0.5 * (pi / a));
	tmp = 0.0;
	if (b <= 3.5e-198)
		tmp = t_1;
	elseif (b <= 7.6e-184)
		tmp = (0.5 / b) * (pi / (a * b));
	elseif (b <= 4.1e-76)
		tmp = t_1;
	else
		tmp = t_0 * (0.5 * (pi / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.5e-198], t$95$1, If[LessEqual[b, 7.6e-184], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.1e-76], t$95$1, N[(t$95$0 * N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.50000000000000025e-198 or 7.60000000000000033e-184 < b < 4.0999999999999998e-76

   1. Initial program 80.3%

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

      1. un-div-inv 80.3%

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

      2. difference-of-squares 89.9%

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

      3. associate-/r* 90.3%

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

      4. div-inv 90.3%

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

      5. metadata-eval 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. associate-*l/ 99.7%

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

      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/ 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. sub-neg 99.7%

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

      6. distribute-neg-frac 99.7%

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

      7. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   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. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around inf 75.5%

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

   if 3.50000000000000025e-198 < b < 7.60000000000000033e-184

   1. Initial program 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*r* 86.4%

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

      3. associate-*r/ 86.7%

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

      4. associate-*r* 86.7%

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

      5. *-rgt-identity 86.7%

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

      6. sub-neg 86.7%

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

      7. distribute-neg-frac 86.7%

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

      8. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. *-un-lft-identity 86.7%

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

      2. difference-of-squares 86.7%

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

      3. times-frac 99.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 14.5%

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

      6. frac-times 14.5%

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

      7. metadata-eval 14.5%

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

      8. metadata-eval 14.5%

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

      9. frac-times 14.5%

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

      10. sqrt-unprod 57.9%

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

      11. add-sqr-sqrt 57.9%

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

      12. div-inv 57.9%

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

      13. metadata-eval 57.9%

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

   6. Applied egg-rr 57.9%

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

      1. associate-*l/ 57.9%

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

      2. *-lft-identity 57.9%

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

      3. associate-/l* 57.9%

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

      4. associate-*l/ 57.9%

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

      5. +-commutative 57.9%

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

      6. +-commutative 57.9%

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

      7. *-commutative 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in b around 0 57.9%

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

       1. associate-*l/ 57.9%

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

       2. *-un-lft-identity 57.9%

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

       3. associate-/l* 57.9%

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

   11. Applied egg-rr 57.9%

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

       1. *-commutative 57.9%

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

       2. times-frac 58.1%

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

   13. Simplified 58.1%

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

   14. Taylor expanded in b around inf 60.2%

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

   if 4.0999999999999998e-76 < b

   1. Initial program 80.5%

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

      1. un-div-inv 80.5%

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

      2. difference-of-squares 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* 88.7%

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

      4. div-inv 88.7%

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

      5. metadata-eval 88.7%

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

   4. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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* 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/l* 99.8%

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

      2. associate-*r/ 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. sub-neg 99.7%

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

      6. distribute-neg-frac 99.7%

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

      7. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 99.6%

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

       1. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around 0 78.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{if}\;b \leq 3.5 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b - a}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ 1.0 a) b) (* 0.5 (/ PI a)))))
   (if (<= b 3.5e-198)
     t_0
     (if (<= b 7.6e-184)
       (* (/ 0.5 b) (/ PI (* a b)))
       (if (<= b 5.8e-81) t_0 (* (/ 0.5 b) (/ (/ PI (- b a)) a)))))))

C

double code(double a, double b) {
	double t_0 = ((1.0 / a) / b) * (0.5 * (((double) M_PI) / a));
	double tmp;
	if (b <= 3.5e-198) {
		tmp = t_0;
	} else if (b <= 7.6e-184) {
		tmp = (0.5 / b) * (((double) M_PI) / (a * b));
	} else if (b <= 5.8e-81) {
		tmp = t_0;
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / (b - a)) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((1.0 / a) / b) * (0.5 * (Math.PI / a));
	double tmp;
	if (b <= 3.5e-198) {
		tmp = t_0;
	} else if (b <= 7.6e-184) {
		tmp = (0.5 / b) * (Math.PI / (a * b));
	} else if (b <= 5.8e-81) {
		tmp = t_0;
	} else {
		tmp = (0.5 / b) * ((Math.PI / (b - a)) / a);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((1.0 / a) / b) * (0.5 * (math.pi / a))
	tmp = 0
	if b <= 3.5e-198:
		tmp = t_0
	elif b <= 7.6e-184:
		tmp = (0.5 / b) * (math.pi / (a * b))
	elif b <= 5.8e-81:
		tmp = t_0
	else:
		tmp = (0.5 / b) * ((math.pi / (b - a)) / a)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(1.0 / a) / b) * Float64(0.5 * Float64(pi / a)))
	tmp = 0.0
	if (b <= 3.5e-198)
		tmp = t_0;
	elseif (b <= 7.6e-184)
		tmp = Float64(Float64(0.5 / b) * Float64(pi / Float64(a * b)));
	elseif (b <= 5.8e-81)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / Float64(b - a)) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((1.0 / a) / b) * (0.5 * (pi / a));
	tmp = 0.0;
	if (b <= 3.5e-198)
		tmp = t_0;
	elseif (b <= 7.6e-184)
		tmp = (0.5 / b) * (pi / (a * b));
	elseif (b <= 5.8e-81)
		tmp = t_0;
	else
		tmp = (0.5 / b) * ((pi / (b - a)) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.5e-198], t$95$0, If[LessEqual[b, 7.6e-184], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-81], t$95$0, N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.50000000000000025e-198 or 7.60000000000000033e-184 < b < 5.79999999999999978e-81

   1. Initial program 80.0%

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

      1. un-div-inv 80.0%

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

      2. difference-of-squares 89.8%

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

      3. associate-/r* 90.2%

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

      4. div-inv 90.2%

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

      5. metadata-eval 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l/ 99.7%

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

      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/ 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. sub-neg 99.7%

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

      6. distribute-neg-frac 99.7%

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

      7. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   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. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around inf 75.2%

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

   if 3.50000000000000025e-198 < b < 7.60000000000000033e-184

   1. Initial program 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*r* 86.4%

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

      3. associate-*r/ 86.7%

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

      4. associate-*r* 86.7%

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

      5. *-rgt-identity 86.7%

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

      6. sub-neg 86.7%

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

      7. distribute-neg-frac 86.7%

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

      8. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. *-un-lft-identity 86.7%

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

      2. difference-of-squares 86.7%

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

      3. times-frac 99.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 14.5%

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

      6. frac-times 14.5%

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

      7. metadata-eval 14.5%

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

      8. metadata-eval 14.5%

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

      9. frac-times 14.5%

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

      10. sqrt-unprod 57.9%

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

      11. add-sqr-sqrt 57.9%

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

      12. div-inv 57.9%

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

      13. metadata-eval 57.9%

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

   6. Applied egg-rr 57.9%

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

      1. associate-*l/ 57.9%

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

      2. *-lft-identity 57.9%

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

      3. associate-/l* 57.9%

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

      4. associate-*l/ 57.9%

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

      5. +-commutative 57.9%

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

      6. +-commutative 57.9%

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

      7. *-commutative 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in b around 0 57.9%

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

       1. associate-*l/ 57.9%

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

       2. *-un-lft-identity 57.9%

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

       3. associate-/l* 57.9%

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

   11. Applied egg-rr 57.9%

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

       1. *-commutative 57.9%

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

       2. times-frac 58.1%

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

   13. Simplified 58.1%

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

   14. Taylor expanded in b around inf 60.2%

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

   if 5.79999999999999978e-81 < b

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r* 81.3%

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

      3. associate-*r/ 81.2%

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

      4. associate-*r* 81.2%

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

      5. *-rgt-identity 81.2%

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

      6. sub-neg 81.2%

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

      7. distribute-neg-frac 81.2%

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

      8. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. difference-of-squares 89.0%

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

      3. times-frac 98.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 88.6%

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

      6. frac-times 88.6%

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

      7. metadata-eval 88.6%

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

      8. metadata-eval 88.6%

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

      9. frac-times 88.6%

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

      10. sqrt-unprod 88.6%

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

      11. add-sqr-sqrt 88.6%

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

      12. div-inv 88.6%

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

      13. metadata-eval 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. associate-*l/ 88.5%

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

      2. *-lft-identity 88.5%

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

      3. associate-/l* 89.9%

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

      4. associate-*l/ 90.0%

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

      5. +-commutative 90.0%

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

      6. +-commutative 90.0%

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

      7. *-commutative 90.0%

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

   8. Simplified 90.0%

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

   9. Taylor expanded in b around 0 89.9%

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

       1. associate-*l/ 90.0%

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

       2. *-un-lft-identity 90.0%

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

       3. associate-/l* 90.0%

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

   11. Applied egg-rr 90.0%

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

       1. *-commutative 90.0%

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

       2. times-frac 90.0%

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

   13. Simplified 90.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{a}}{b} \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b - a}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+114}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{-\pi}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3e+114)
   (* (/ 0.5 b) (/ (/ (- PI) a) a))
   (* (/ 0.5 b) (/ PI (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3e+114) {
		tmp = (0.5 / b) * ((-((double) M_PI) / a) / a);
	} else {
		tmp = (0.5 / b) * (((double) M_PI) / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3e+114) {
		tmp = (0.5 / b) * ((-Math.PI / a) / a);
	} else {
		tmp = (0.5 / b) * (Math.PI / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3e+114:
		tmp = (0.5 / b) * ((-math.pi / a) / a)
	else:
		tmp = (0.5 / b) * (math.pi / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3e+114)
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(Float64(-pi) / a) / a));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(pi / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3e+114)
		tmp = (0.5 / b) * ((-pi / a) / a);
	else
		tmp = (0.5 / b) * (pi / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3e+114], N[(N[(0.5 / b), $MachinePrecision] * N[(N[((-Pi) / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3e114

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-*r* 51.7%

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

      3. associate-*r/ 51.7%

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

      4. associate-*r* 51.7%

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

      5. *-rgt-identity 51.7%

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

      6. sub-neg 51.7%

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

      7. distribute-neg-frac 51.7%

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

      8. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

      1. *-un-lft-identity 51.7%

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

      2. difference-of-squares 78.1%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 60.3%

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

      5. sqrt-unprod 70.5%

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

      6. frac-times 70.5%

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

      7. metadata-eval 70.5%

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

      8. metadata-eval 70.5%

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

      9. frac-times 70.5%

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

      10. sqrt-unprod 23.2%

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

      11. add-sqr-sqrt 71.3%

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

      12. div-inv 71.3%

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

      13. metadata-eval 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. associate-*l/ 71.3%

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

      2. *-lft-identity 71.3%

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

      3. associate-/l* 71.3%

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

      4. associate-*l/ 71.3%

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

      5. +-commutative 71.3%

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

      6. +-commutative 71.3%

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

      7. *-commutative 71.3%

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

   8. Simplified 71.3%

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

   9. Taylor expanded in b around 0 71.3%

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

       1. associate-*l/ 71.3%

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

       2. *-un-lft-identity 71.3%

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

       3. associate-/l* 71.3%

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

   11. Applied egg-rr 71.3%

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

       1. *-commutative 71.3%

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

       2. times-frac 71.7%

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

   13. Simplified 71.7%

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

   14. Taylor expanded in b around 0 71.7%

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

       1. associate-*r/ 71.7%

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

       2. mul-1-neg 71.7%

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

   16. Simplified 71.7%

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

   if -3e114 < a

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-*r* 85.5%

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

      3. associate-*r/ 85.5%

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

      4. associate-*r* 85.5%

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

      5. *-rgt-identity 85.5%

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

      6. sub-neg 85.5%

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

      7. distribute-neg-frac 85.5%

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

      8. metadata-eval 85.5%

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

   3. Simplified 85.5%

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

      1. *-un-lft-identity 85.5%

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

      2. difference-of-squares 91.5%

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

      3. times-frac 99.2%

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

      4. add-sqr-sqrt 55.7%

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

      5. sqrt-unprod 73.3%

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

      6. frac-times 73.2%

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

      7. metadata-eval 73.2%

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

      8. metadata-eval 73.2%

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

      9. frac-times 73.3%

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

      10. sqrt-unprod 26.9%

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

      11. add-sqr-sqrt 58.8%

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

      12. div-inv 58.8%

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

      13. metadata-eval 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. associate-*l/ 58.8%

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

      2. *-lft-identity 58.8%

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

      3. associate-/l* 59.2%

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

      4. associate-*l/ 59.3%

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

      5. +-commutative 59.3%

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

      6. +-commutative 59.3%

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

      7. *-commutative 59.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in b around 0 59.2%

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

       1. associate-*l/ 59.2%

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

       2. *-un-lft-identity 59.2%

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

       3. associate-/l* 59.2%

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

   11. Applied egg-rr 59.2%

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

       1. *-commutative 59.2%

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

       2. times-frac 59.3%

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

   13. Simplified 59.3%

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

   14. Taylor expanded in b around inf 58.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+114}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{-\pi}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. un-div-inv 80.6%

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

   2. difference-of-squares 89.5%

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

   3. associate-/r* 89.8%

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

   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. associate-*l/ 99.7%

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

   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* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. *-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. metadata-eval 99.7%

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

8. Simplified 99.7%

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

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. Final simplification 99.7%

    \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{1}{a \cdot b} \]
11. Add Preprocessing

Alternative 7: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. un-div-inv 80.6%

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

   2. difference-of-squares 89.5%

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

   3. associate-/r* 89.8%

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

   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. associate-*l/ 99.7%

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

   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* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. *-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. metadata-eval 99.7%

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

8. Simplified 99.7%

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

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. associate-/r* 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{0.5 \cdot \pi}{a + b} \cdot \frac{\frac{1}{a}}{b} \]
13. Add Preprocessing

Alternative 8: 99.2% 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 80.5%

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

   1. un-div-inv 80.6%

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

   2. difference-of-squares 89.5%

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

   3. associate-/r* 89.8%

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

   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. associate-*l/ 99.7%

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

   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* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. *-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. metadata-eval 99.7%

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

8. Simplified 99.7%

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

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. associate-/r* 99.7%

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

11. Simplified 99.7%

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

    1. *-commutative 99.7%

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

    2. associate-/l/ 99.7%

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

    3. frac-times 99.2%

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

    4. *-un-lft-identity 99.2%

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

    5. *-commutative 99.2%

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

13. Applied egg-rr 99.2%

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

14. Final simplification 99.2%

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

Alternative 9: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. un-div-inv 80.6%

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

   2. difference-of-squares 89.5%

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

   3. associate-/r* 89.8%

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

   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. associate-*l/ 99.7%

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

   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* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. *-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. metadata-eval 99.7%

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

8. Simplified 99.7%

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

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. associate-/r* 99.7%

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

11. Simplified 99.7%

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

    1. associate-*l/ 99.6%

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

    2. associate-/l/ 99.6%

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

    3. div-inv 99.7%

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

    4. times-frac 99.3%

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

    5. clear-num 99.3%

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

    6. frac-times 99.6%

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

    7. metadata-eval 99.6%

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

13. Applied egg-rr 99.6%

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

    1. associate-/r* 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 10: 63.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. *-commutative 80.5%

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

   2. associate-*r* 80.5%

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

   3. associate-*r/ 80.5%

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

   4. associate-*r* 80.5%

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

   5. *-rgt-identity 80.5%

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

   6. sub-neg 80.5%

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

   7. distribute-neg-frac 80.5%

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

   8. metadata-eval 80.5%

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

3. Simplified 80.5%

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

   1. *-un-lft-identity 80.5%

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

   2. difference-of-squares 89.5%

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

   3. times-frac 99.3%

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

   4. add-sqr-sqrt 56.4%

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

   5. sqrt-unprod 72.9%

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

   6. frac-times 72.8%

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

   7. metadata-eval 72.8%

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

   8. metadata-eval 72.8%

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

   9. frac-times 72.9%

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

   10. sqrt-unprod 26.3%

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

   11. add-sqr-sqrt 60.7%

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

   12. div-inv 60.7%

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

   13. metadata-eval 60.7%

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

6. Applied egg-rr 60.7%

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

   1. associate-*l/ 60.6%

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

   2. *-lft-identity 60.6%

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

   3. associate-/l* 61.0%

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

   4. associate-*l/ 61.1%

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

   5. +-commutative 61.1%

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

   6. +-commutative 61.1%

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

   7. *-commutative 61.1%

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

8. Simplified 61.1%

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

9. Taylor expanded in b around 0 61.0%

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

    1. associate-*l/ 61.0%

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

    2. *-un-lft-identity 61.0%

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

    3. associate-/l* 61.0%

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

11. Applied egg-rr 61.0%

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

    1. *-commutative 61.0%

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

    2. times-frac 61.1%

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

13. Simplified 61.1%

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

14. Taylor expanded in b around inf 56.0%

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

15. Final simplification 56.0%

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

Reproduce

herbie shell --seed 2024079 
(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))))