NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.6%

Time: 11.7s

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: 78.1% 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, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-*l* 77.2%

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

   2. *-rgt-identity 77.2%

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

   3. associate-/l* 77.2%

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

   4. metadata-eval 77.2%

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

   5. associate-*l/ 77.3%

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

   6. *-lft-identity 77.3%

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

   7. sub-neg 77.3%

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

   8. distribute-neg-frac 77.3%

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

   9. metadata-eval 77.3%

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

3. Simplified 77.3%

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

   1. *-un-lft-identity 77.3%

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

   2. clear-num 76.7%

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

   3. frac-add 76.7%

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

   4. associate-/r/ 76.7%

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

   5. *-un-lft-identity 76.7%

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

   6. *-commutative 76.7%

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

   7. neg-mul-1 76.7%

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

   8. sub-neg 76.7%

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

   9. flip-+ 98.5%

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

   10. +-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

   1. *-lft-identity 98.5%

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

   2. associate-/r* 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+99}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e+99)
   (* PI (/ 0.5 (* a (* b (+ a b)))))
   (/ (/ (* 0.5 (/ PI b)) a) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e+99) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = ((0.5 * (((double) M_PI) / b)) / a) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.8e+99) {
		tmp = Math.PI * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = ((0.5 * (Math.PI / b)) / a) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.8e+99:
		tmp = math.pi * (0.5 / (a * (b * (a + b))))
	else:
		tmp = ((0.5 * (math.pi / b)) / a) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e+99)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * Float64(a + b)))));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(pi / b)) / a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.8e+99)
		tmp = pi * (0.5 / (a * (b * (a + b))));
	else
		tmp = ((0.5 * (pi / b)) / a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e+99], N[(Pi * N[(0.5 / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000004e99

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*r* 82.7%

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

      3. associate-*r/ 82.7%

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

      4. associate-*r* 82.7%

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

      5. *-rgt-identity 82.7%

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

      6. sub-neg 82.7%

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

      7. distribute-neg-frac 82.7%

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

      8. metadata-eval 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 82.7%

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

      2. div-inv 82.7%

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

      3. metadata-eval 82.7%

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

      4. *-commutative 82.7%

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

      5. associate-*r* 82.7%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in a around 0 90.8%

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

      1. unpow2 90.8%

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

      2. distribute-rgt-in 95.2%

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

   9. Simplified 95.2%

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

   if 5.8000000000000004e99 < b

   1. Initial program 54.8%

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

      1. associate-*l* 54.8%

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

      2. *-rgt-identity 54.8%

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

      3. associate-/l* 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*l/ 54.8%

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

      6. *-lft-identity 54.8%

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

      7. sub-neg 54.8%

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

      8. distribute-neg-frac 54.8%

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

      9. metadata-eval 54.8%

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

   3. Simplified 54.8%

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

      1. metadata-eval 54.8%

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

      2. div-inv 54.8%

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

      3. clear-num 54.8%

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

      4. clear-num 53.2%

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

      5. frac-times 53.3%

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

      6. metadata-eval 53.3%

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

      7. frac-add 53.3%

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

      8. associate-/r/ 53.3%

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

      9. *-un-lft-identity 53.3%

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

      10. *-commutative 53.3%

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

      11. neg-mul-1 53.3%

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

      12. sub-neg 53.3%

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

      13. flip-+ 97.2%

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

      14. +-commutative 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. associate-/l/ 97.2%

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

      2. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. clear-num 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/l/ 97.2%

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

      5. *-commutative 97.2%

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

      6. div-inv 97.3%

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

      7. div-inv 97.3%

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

      8. metadata-eval 97.3%

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

      9. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around 0 99.6%

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

       1. associate-*r/ 99.7%

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

       2. associate-/r* 99.8%

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

       3. *-commutative 99.8%

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

   13. Applied egg-rr 99.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+99}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{-80}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= a -1.65e-80) (* t_0 (/ PI a)) (* t_0 (/ PI b)))))

C

double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (a <= -1.65e-80) {
		tmp = t_0 * (((double) M_PI) / a);
	} else {
		tmp = t_0 * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (a <= -1.65e-80) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = t_0 * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 / (a * b)
	tmp = 0
	if a <= -1.65e-80:
		tmp = t_0 * (math.pi / a)
	else:
		tmp = t_0 * (math.pi / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 / Float64(a * b))
	tmp = 0.0
	if (a <= -1.65e-80)
		tmp = Float64(t_0 * Float64(pi / a));
	else
		tmp = Float64(t_0 * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 / (a * b);
	tmp = 0.0;
	if (a <= -1.65e-80)
		tmp = t_0 * (pi / a);
	else
		tmp = t_0 * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.65e-80], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.65e-80

   1. Initial program 80.2%

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

      1. associate-*l* 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. associate-/l* 80.1%

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

      4. metadata-eval 80.1%

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

      5. associate-*l/ 80.1%

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

      6. *-lft-identity 80.1%

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

      7. sub-neg 80.1%

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

      8. distribute-neg-frac 80.1%

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

      9. metadata-eval 80.1%

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

   3. Simplified 80.1%

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

      1. metadata-eval 80.1%

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

      2. div-inv 80.1%

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

      3. clear-num 80.1%

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

      4. clear-num 79.1%

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

      5. frac-times 79.0%

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

      6. metadata-eval 79.0%

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

      7. frac-add 79.0%

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

      8. associate-/r/ 79.1%

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

      9. *-un-lft-identity 79.1%

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

      10. *-commutative 79.1%

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

      11. neg-mul-1 79.1%

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

      12. sub-neg 79.1%

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

      13. flip-+ 97.8%

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

      14. +-commutative 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-/l/ 97.9%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. clear-num 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/l/ 97.9%

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

      5. *-commutative 97.9%

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

      6. div-inv 98.0%

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

      7. div-inv 98.0%

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

      8. metadata-eval 98.0%

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

      9. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in a around inf 87.6%

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

   if -1.65e-80 < a

   1. Initial program 76.1%

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

      1. associate-*l* 76.1%

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

      2. *-rgt-identity 76.1%

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

      3. associate-/l* 76.1%

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

      4. metadata-eval 76.1%

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

      5. associate-*l/ 76.2%

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

      6. *-lft-identity 76.2%

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

      7. sub-neg 76.2%

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

      8. distribute-neg-frac 76.2%

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

      9. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

      1. metadata-eval 76.2%

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

      2. div-inv 76.2%

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

      3. clear-num 76.2%

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

      4. clear-num 75.7%

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

      5. frac-times 75.7%

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

      6. metadata-eval 75.7%

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

      7. frac-add 75.7%

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

      8. associate-/r/ 75.7%

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

      9. *-un-lft-identity 75.7%

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

      10. *-commutative 75.7%

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

      11. neg-mul-1 75.7%

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

      12. sub-neg 75.7%

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

      13. flip-+ 98.6%

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

      14. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. div-inv 99.7%

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

      2. clear-num 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/l/ 98.7%

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

      5. *-commutative 98.7%

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

      6. div-inv 98.7%

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

      7. div-inv 98.7%

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

      8. metadata-eval 98.7%

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

      9. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around 0 75.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-80}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.8e-62)
   (* (/ 0.5 (* a b)) (/ PI a))
   (* PI (/ (/ 0.5 b) (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-62) {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / a);
	} else {
		tmp = ((double) M_PI) * ((0.5 / b) / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-62) {
		tmp = (0.5 / (a * b)) * (Math.PI / a);
	} else {
		tmp = Math.PI * ((0.5 / b) / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.8e-62:
		tmp = (0.5 / (a * b)) * (math.pi / a)
	else:
		tmp = math.pi * ((0.5 / b) / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.8e-62)
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / a));
	else
		tmp = Float64(pi * Float64(Float64(0.5 / b) / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.8e-62)
		tmp = (0.5 / (a * b)) * (pi / a);
	else
		tmp = pi * ((0.5 / b) / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.8e-62], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(0.5 / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.80000000000000002e-62

   1. Initial program 79.3%

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

      1. associate-*l* 79.2%

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

      2. *-rgt-identity 79.2%

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

      3. associate-/l* 79.2%

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

      4. metadata-eval 79.2%

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

      5. associate-*l/ 79.3%

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

      6. *-lft-identity 79.3%

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

      7. sub-neg 79.3%

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

      8. distribute-neg-frac 79.3%

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

      9. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

      1. metadata-eval 79.3%

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

      2. div-inv 79.3%

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

      3. clear-num 79.3%

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

      4. clear-num 78.2%

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

      5. frac-times 78.1%

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

      6. metadata-eval 78.1%

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

      7. frac-add 78.1%

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

      8. associate-/r/ 78.2%

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

      9. *-un-lft-identity 78.2%

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

      10. *-commutative 78.2%

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

      11. neg-mul-1 78.2%

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

      12. sub-neg 78.2%

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

      13. flip-+ 97.7%

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

      14. +-commutative 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. associate-/l/ 97.8%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. clear-num 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/l/ 97.8%

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

      5. *-commutative 97.8%

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

      6. div-inv 98.0%

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

      7. div-inv 98.0%

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

      8. metadata-eval 98.0%

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

      9. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in a around inf 87.0%

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

   if -2.80000000000000002e-62 < a

   1. Initial program 76.5%

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

      1. associate-*l* 76.5%

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

      2. *-rgt-identity 76.5%

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

      3. associate-/l* 76.5%

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

      4. metadata-eval 76.5%

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

      5. associate-*l/ 76.5%

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

      6. *-lft-identity 76.5%

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

      7. sub-neg 76.5%

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

      8. distribute-neg-frac 76.5%

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

      9. metadata-eval 76.5%

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

   3. Simplified 76.5%

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

      1. metadata-eval 76.5%

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

      2. div-inv 76.5%

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

      3. clear-num 76.5%

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

      4. clear-num 76.1%

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

      5. frac-times 76.1%

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

      6. metadata-eval 76.1%

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

      7. frac-add 76.1%

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

      8. associate-/r/ 76.1%

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

      9. *-un-lft-identity 76.1%

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

      10. *-commutative 76.1%

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

      11. neg-mul-1 76.1%

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

      12. sub-neg 76.1%

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

      13. flip-+ 98.7%

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

      14. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. div-inv 99.7%

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

      2. clear-num 99.7%

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

      3. *-commutative 99.7%

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

      4. div-inv 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-*l* 99.7%

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

      7. associate-/l/ 98.7%

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

      8. *-commutative 98.7%

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

      9. div-inv 98.7%

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

      10. *-commutative 98.7%

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

      11. associate-/r* 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around 0 75.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.8e-62)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* PI (/ (/ 0.5 b) (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-62) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((double) M_PI) * ((0.5 / b) / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-62) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = Math.PI * ((0.5 / b) / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.8e-62:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = math.pi * ((0.5 / b) / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.8e-62)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(pi * Float64(Float64(0.5 / b) / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.8e-62)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = pi * ((0.5 / b) / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.8e-62], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(0.5 / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.80000000000000002e-62

   1. Initial program 79.3%

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

      1. associate-*l* 79.2%

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

      2. *-rgt-identity 79.2%

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

      3. associate-/l* 79.2%

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

      4. metadata-eval 79.2%

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

      5. associate-*l/ 79.3%

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

      6. *-lft-identity 79.3%

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

      7. sub-neg 79.3%

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

      8. distribute-neg-frac 79.3%

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

      9. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. clear-num 78.2%

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

      3. frac-add 78.2%

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

      4. associate-/r/ 78.3%

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

      5. *-un-lft-identity 78.3%

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

      6. *-commutative 78.3%

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

      7. neg-mul-1 78.3%

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

      8. sub-neg 78.3%

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

      9. flip-+ 97.8%

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

      10. +-commutative 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. metadata-eval 99.6%

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

      2. div-inv 99.6%

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

      3. clear-num 99.6%

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

      4. associate-/l/ 97.8%

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

      5. *-commutative 97.8%

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

      6. frac-times 97.7%

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

      7. metadata-eval 97.7%

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

   10. Applied egg-rr 97.7%

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

       1. *-commutative 97.7%

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

       2. associate-/r* 97.8%

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

       3. associate-/r* 99.6%

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

       4. associate-/r/ 99.6%

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

       5. associate-/r* 99.6%

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

       6. *-commutative 99.6%

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

       7. associate-/r* 99.6%

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

       8. associate-/l/ 99.6%

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

       9. associate-/r* 99.6%

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

       10. metadata-eval 99.6%

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

       11. *-commutative 99.6%

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

       12. associate-*r/ 99.7%

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

       13. associate-/l* 99.8%

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

       14. associate-/l/ 98.0%

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

       15. distribute-lft-out 83.7%

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

       16. associate-*l* 83.7%

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

       17. unpow2 83.7%

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

       18. *-commutative 83.7%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in b around 0 87.2%

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

       1. *-commutative 85.2%

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

   15. Simplified 87.2%

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

   if -2.80000000000000002e-62 < a

   1. Initial program 76.5%

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

      1. associate-*l* 76.5%

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

      2. *-rgt-identity 76.5%

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

      3. associate-/l* 76.5%

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

      4. metadata-eval 76.5%

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

      5. associate-*l/ 76.5%

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

      6. *-lft-identity 76.5%

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

      7. sub-neg 76.5%

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

      8. distribute-neg-frac 76.5%

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

      9. metadata-eval 76.5%

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

   3. Simplified 76.5%

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

      1. metadata-eval 76.5%

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

      2. div-inv 76.5%

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

      3. clear-num 76.5%

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

      4. clear-num 76.1%

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

      5. frac-times 76.1%

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

      6. metadata-eval 76.1%

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

      7. frac-add 76.1%

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

      8. associate-/r/ 76.1%

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

      9. *-un-lft-identity 76.1%

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

      10. *-commutative 76.1%

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

      11. neg-mul-1 76.1%

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

      12. sub-neg 76.1%

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

      13. flip-+ 98.7%

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

      14. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. div-inv 99.7%

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

      2. clear-num 99.7%

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

      3. *-commutative 99.7%

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

      4. div-inv 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-*l* 99.7%

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

      7. associate-/l/ 98.7%

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

      8. *-commutative 98.7%

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

      9. div-inv 98.7%

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

      10. *-commutative 98.7%

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

      11. associate-/r* 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around 0 75.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.8e-62)
   (/ (* 0.5 (/ PI a)) (* a b))
   (/ (/ (* 0.5 (/ PI b)) a) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-62) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((0.5 * (((double) M_PI) / b)) / a) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-62) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((0.5 * (Math.PI / b)) / a) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.8e-62:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((0.5 * (math.pi / b)) / a) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.8e-62)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(pi / b)) / a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.8e-62)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((0.5 * (pi / b)) / a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.8e-62], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.80000000000000002e-62

   1. Initial program 79.3%

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

      1. associate-*l* 79.2%

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

      2. *-rgt-identity 79.2%

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

      3. associate-/l* 79.2%

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

      4. metadata-eval 79.2%

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

      5. associate-*l/ 79.3%

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

      6. *-lft-identity 79.3%

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

      7. sub-neg 79.3%

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

      8. distribute-neg-frac 79.3%

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

      9. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. clear-num 78.2%

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

      3. frac-add 78.2%

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

      4. associate-/r/ 78.3%

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

      5. *-un-lft-identity 78.3%

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

      6. *-commutative 78.3%

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

      7. neg-mul-1 78.3%

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

      8. sub-neg 78.3%

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

      9. flip-+ 97.8%

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

      10. +-commutative 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. metadata-eval 99.6%

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

      2. div-inv 99.6%

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

      3. clear-num 99.6%

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

      4. associate-/l/ 97.8%

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

      5. *-commutative 97.8%

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

      6. frac-times 97.7%

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

      7. metadata-eval 97.7%

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

   10. Applied egg-rr 97.7%

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

       1. *-commutative 97.7%

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

       2. associate-/r* 97.8%

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

       3. associate-/r* 99.6%

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

       4. associate-/r/ 99.6%

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

       5. associate-/r* 99.6%

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

       6. *-commutative 99.6%

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

       7. associate-/r* 99.6%

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

       8. associate-/l/ 99.6%

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

       9. associate-/r* 99.6%

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

       10. metadata-eval 99.6%

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

       11. *-commutative 99.6%

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

       12. associate-*r/ 99.7%

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

       13. associate-/l* 99.8%

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

       14. associate-/l/ 98.0%

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

       15. distribute-lft-out 83.7%

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

       16. associate-*l* 83.7%

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

       17. unpow2 83.7%

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

       18. *-commutative 83.7%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in b around 0 87.2%

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

       1. *-commutative 85.2%

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

   15. Simplified 87.2%

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

   if -2.80000000000000002e-62 < a

   1. Initial program 76.5%

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

      1. associate-*l* 76.5%

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

      2. *-rgt-identity 76.5%

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

      3. associate-/l* 76.5%

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

      4. metadata-eval 76.5%

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

      5. associate-*l/ 76.5%

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

      6. *-lft-identity 76.5%

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

      7. sub-neg 76.5%

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

      8. distribute-neg-frac 76.5%

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

      9. metadata-eval 76.5%

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

   3. Simplified 76.5%

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

      1. metadata-eval 76.5%

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

      2. div-inv 76.5%

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

      3. clear-num 76.5%

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

      4. clear-num 76.1%

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

      5. frac-times 76.1%

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

      6. metadata-eval 76.1%

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

      7. frac-add 76.1%

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

      8. associate-/r/ 76.1%

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

      9. *-un-lft-identity 76.1%

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

      10. *-commutative 76.1%

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

      11. neg-mul-1 76.1%

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

      12. sub-neg 76.1%

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

      13. flip-+ 98.7%

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

      14. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. associate-/l/ 98.6%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. div-inv 99.7%

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

      2. clear-num 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/l/ 98.7%

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

      5. *-commutative 98.7%

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

      6. div-inv 98.7%

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

      7. div-inv 98.7%

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

      8. metadata-eval 98.7%

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

      9. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around 0 75.3%

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

       1. associate-*r/ 75.3%

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

       2. associate-/r* 75.3%

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

       3. *-commutative 75.3%

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

   13. Applied egg-rr 75.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r* 77.2%

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

   3. associate-*r/ 77.3%

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

   4. associate-*r* 77.3%

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

   5. *-rgt-identity 77.3%

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

   6. sub-neg 77.3%

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

   7. distribute-neg-frac 77.3%

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

   8. metadata-eval 77.3%

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

3. Simplified 77.3%

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

   1. associate-*l/ 77.3%

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

   2. div-inv 77.3%

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

   3. metadata-eval 77.3%

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

   4. *-commutative 77.3%

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

   5. associate-*r* 77.3%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 8: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-*l* 77.2%

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

   2. *-rgt-identity 77.2%

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

   3. associate-/l* 77.2%

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

   4. metadata-eval 77.2%

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

   5. associate-*l/ 77.3%

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

   6. *-lft-identity 77.3%

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

   7. sub-neg 77.3%

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

   8. distribute-neg-frac 77.3%

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

   9. metadata-eval 77.3%

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

3. Simplified 77.3%

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

   1. metadata-eval 77.3%

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

   2. div-inv 77.3%

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

   3. clear-num 77.3%

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

   4. clear-num 76.7%

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

   5. frac-times 76.7%

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

   6. metadata-eval 76.7%

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

   7. frac-add 76.6%

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

   8. associate-/r/ 76.7%

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

   9. *-un-lft-identity 76.7%

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

   10. *-commutative 76.7%

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

   11. neg-mul-1 76.7%

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

   12. sub-neg 76.7%

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

   13. flip-+ 98.4%

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

   14. +-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-/l/ 98.4%

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

   2. associate-/r* 99.6%

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

8. Simplified 99.6%

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

   1. div-inv 99.7%

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

   2. clear-num 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. div-inv 98.5%

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

   7. div-inv 98.5%

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

   8. metadata-eval 98.5%

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

   9. times-frac 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 9: 61.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r* 77.2%

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

   3. associate-*r/ 77.3%

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

   4. associate-*r* 77.3%

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

   5. *-rgt-identity 77.3%

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

   6. sub-neg 77.3%

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

   7. distribute-neg-frac 77.3%

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

   8. metadata-eval 77.3%

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

3. Simplified 77.3%

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

   1. associate-*l/ 77.3%

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

   2. div-inv 77.3%

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

   3. metadata-eval 77.3%

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

   4. *-commutative 77.3%

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

   5. associate-*r* 77.3%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in a around 0 83.9%

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

   1. unpow2 83.9%

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

   2. distribute-rgt-in 89.7%

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

9. Simplified 89.7%

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

10. Taylor expanded in b around 0 60.8%

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

    1. *-commutative 60.8%

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

12. Simplified 60.8%

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

13. Final simplification 60.8%

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

Alternative 10: 61.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-*l* 77.2%

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

   2. *-rgt-identity 77.2%

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

   3. associate-/l* 77.2%

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

   4. metadata-eval 77.2%

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

   5. associate-*l/ 77.3%

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

   6. *-lft-identity 77.3%

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

   7. sub-neg 77.3%

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

   8. distribute-neg-frac 77.3%

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

   9. metadata-eval 77.3%

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

3. Simplified 77.3%

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

   1. metadata-eval 77.3%

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

   2. div-inv 77.3%

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

   3. clear-num 77.3%

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

   4. clear-num 76.7%

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

   5. frac-times 76.7%

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

   6. metadata-eval 76.7%

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

   7. frac-add 76.6%

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

   8. associate-/r/ 76.7%

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

   9. *-un-lft-identity 76.7%

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

   10. *-commutative 76.7%

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

   11. neg-mul-1 76.7%

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

   12. sub-neg 76.7%

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

   13. flip-+ 98.4%

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

   14. +-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-/l/ 98.4%

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

   2. associate-/r* 99.6%

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

8. Simplified 99.6%

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

   1. div-inv 99.7%

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

   2. clear-num 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. div-inv 98.5%

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

   7. div-inv 98.5%

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

   8. metadata-eval 98.5%

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

   9. times-frac 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in a around inf 61.3%

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

12. Final simplification 61.3%

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

Reproduce

herbie shell --seed 2024130 
(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))))