NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.5% → 99.6%

Time: 9.7s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

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

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 78.3%

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

   1. associate-*l* 78.3%

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

   2. *-rgt-identity 78.3%

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

   3. associate-/l* 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l/ 78.3%

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

   6. *-lft-identity 78.3%

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

   7. sub-neg 78.3%

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

   8. distribute-neg-frac 78.3%

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

   9. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. *-commutative 78.3%

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

   4. clear-num 78.1%

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

   5. frac-times 78.1%

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

   6. *-un-lft-identity 78.1%

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

   7. frac-add 78.1%

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

   8. associate-/r/ 78.2%

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

   9. *-un-lft-identity 78.2%

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

   10. *-commutative 78.2%

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

   11. neg-mul-1 78.2%

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

   12. sub-neg 78.2%

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

   13. flip-+ 99.0%

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

   14. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-/r* 99.0%

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

8. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. metadata-eval 99.0%

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

   3. 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. Add Preprocessing

Alternative 2: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.3e-76)
   (* (/ 0.5 (* a b)) (/ PI a))
   (* (/ PI (* a b)) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.3e-76) {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / a);
	} else {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.3e-76) {
		tmp = (0.5 / (a * b)) * (Math.PI / a);
	} else {
		tmp = (Math.PI / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.3e-76:
		tmp = (0.5 / (a * b)) * (math.pi / a)
	else:
		tmp = (math.pi / (a * b)) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.3e-76)
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / a));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.3e-76)
		tmp = (0.5 / (a * b)) * (pi / a);
	else
		tmp = (pi / (a * b)) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.3e-76], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.3e-76

   1. Initial program 79.5%

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

      1. associate-*l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/l* 79.5%

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

      4. metadata-eval 79.5%

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

      5. associate-*l/ 79.5%

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

      6. *-lft-identity 79.5%

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

      7. sub-neg 79.5%

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

      8. distribute-neg-frac 79.5%

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

      9. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

      1. metadata-eval 79.5%

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

      2. div-inv 79.5%

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

      3. *-commutative 79.5%

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

      4. clear-num 78.8%

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

      5. frac-times 79.0%

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

      6. *-un-lft-identity 79.0%

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

      7. frac-add 78.9%

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

      8. associate-/r/ 79.0%

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

      9. *-un-lft-identity 79.0%

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

      10. *-commutative 79.0%

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

      11. neg-mul-1 79.0%

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

      12. sub-neg 79.0%

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

      13. flip-+ 99.1%

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

      14. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/r* 99.1%

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

   8. Simplified 99.1%

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

      1. div-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. 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 85.8%

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

   if -5.3e-76 < a

   1. Initial program 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-rgt-identity 77.8%

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

      3. associate-/l* 77.8%

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

      4. metadata-eval 77.8%

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

      5. associate-*l/ 77.8%

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

      6. *-lft-identity 77.8%

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

      7. sub-neg 77.8%

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

      8. distribute-neg-frac 77.8%

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

      9. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

      1. metadata-eval 77.8%

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

      2. div-inv 77.8%

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

      3. *-commutative 77.8%

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

      4. clear-num 77.8%

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

      5. frac-times 77.8%

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

      6. *-un-lft-identity 77.8%

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

      7. frac-add 77.7%

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

      8. associate-/r/ 77.8%

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

      9. *-un-lft-identity 77.8%

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

      10. *-commutative 77.8%

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

      11. neg-mul-1 77.8%

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

      12. sub-neg 77.8%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*r* 93.2%

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

      4. times-frac 94.0%

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

   10. Applied egg-rr 94.0%

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

   11. Taylor expanded in a around 0 71.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-74}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= a -3.8e-74) (* 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 <= -3.8e-74) {
		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 <= -3.8e-74) {
		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 <= -3.8e-74:
		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 <= -3.8e-74)
		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 <= -3.8e-74)
		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, -3.8e-74], 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 < -3.7999999999999996e-74

   1. Initial program 79.5%

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

      1. associate-*l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/l* 79.5%

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

      4. metadata-eval 79.5%

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

      5. associate-*l/ 79.5%

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

      6. *-lft-identity 79.5%

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

      7. sub-neg 79.5%

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

      8. distribute-neg-frac 79.5%

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

      9. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

      1. metadata-eval 79.5%

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

      2. div-inv 79.5%

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

      3. *-commutative 79.5%

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

      4. clear-num 78.8%

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

      5. frac-times 79.0%

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

      6. *-un-lft-identity 79.0%

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

      7. frac-add 78.9%

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

      8. associate-/r/ 79.0%

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

      9. *-un-lft-identity 79.0%

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

      10. *-commutative 79.0%

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

      11. neg-mul-1 79.0%

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

      12. sub-neg 79.0%

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

      13. flip-+ 99.1%

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

      14. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/r* 99.1%

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

   8. Simplified 99.1%

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

      1. div-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. 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 85.8%

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

   if -3.7999999999999996e-74 < a

   1. Initial program 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-rgt-identity 77.8%

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

      3. associate-/l* 77.8%

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

      4. metadata-eval 77.8%

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

      5. associate-*l/ 77.8%

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

      6. *-lft-identity 77.8%

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

      7. sub-neg 77.8%

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

      8. distribute-neg-frac 77.8%

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

      9. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

      1. metadata-eval 77.8%

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

      2. div-inv 77.8%

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

      3. *-commutative 77.8%

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

      4. clear-num 77.8%

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

      5. frac-times 77.8%

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

      6. *-un-lft-identity 77.8%

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

      7. frac-add 77.7%

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

      8. associate-/r/ 77.8%

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

      9. *-un-lft-identity 77.8%

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

      10. *-commutative 77.8%

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

      11. neg-mul-1 77.8%

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

      12. sub-neg 77.8%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. 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 0 71.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-74}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.35e-74)
   (* (/ 0.5 (* a b)) (/ PI a))
   (* (/ (/ PI a) b) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.35e-74) {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / a);
	} else {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.35e-74) {
		tmp = (0.5 / (a * b)) * (Math.PI / a);
	} else {
		tmp = ((Math.PI / a) / b) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.35e-74:
		tmp = (0.5 / (a * b)) * (math.pi / a)
	else:
		tmp = ((math.pi / a) / b) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.35e-74)
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / a));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.35e-74)
		tmp = (0.5 / (a * b)) * (pi / a);
	else
		tmp = ((pi / a) / b) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.35e-74], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35000000000000009e-74

   1. Initial program 79.5%

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

      1. associate-*l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/l* 79.5%

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

      4. metadata-eval 79.5%

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

      5. associate-*l/ 79.5%

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

      6. *-lft-identity 79.5%

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

      7. sub-neg 79.5%

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

      8. distribute-neg-frac 79.5%

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

      9. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

      1. metadata-eval 79.5%

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

      2. div-inv 79.5%

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

      3. *-commutative 79.5%

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

      4. clear-num 78.8%

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

      5. frac-times 79.0%

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

      6. *-un-lft-identity 79.0%

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

      7. frac-add 78.9%

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

      8. associate-/r/ 79.0%

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

      9. *-un-lft-identity 79.0%

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

      10. *-commutative 79.0%

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

      11. neg-mul-1 79.0%

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

      12. sub-neg 79.0%

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

      13. flip-+ 99.1%

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

      14. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/r* 99.1%

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

   8. Simplified 99.1%

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

      1. div-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. 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 85.8%

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

   if -1.35000000000000009e-74 < a

   1. Initial program 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-rgt-identity 77.8%

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

      3. associate-/l* 77.8%

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

      4. metadata-eval 77.8%

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

      5. associate-*l/ 77.8%

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

      6. *-lft-identity 77.8%

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

      7. sub-neg 77.8%

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

      8. distribute-neg-frac 77.8%

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

      9. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

      1. metadata-eval 77.8%

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

      2. div-inv 77.8%

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

      3. *-commutative 77.8%

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

      4. clear-num 77.8%

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

      5. frac-times 77.8%

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

      6. *-un-lft-identity 77.8%

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

      7. frac-add 77.7%

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

      8. associate-/r/ 77.8%

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

      9. *-un-lft-identity 77.8%

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

      10. *-commutative 77.8%

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

      11. neg-mul-1 77.8%

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

      12. sub-neg 77.8%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l/ 98.9%

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

      5. *-commutative 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. associate-*r/ 98.9%

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

       2. associate-/r* 99.7%

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

       3. associate-/l* 99.6%

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

       4. associate-*l/ 99.7%

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

       5. *-commutative 99.7%

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

       6. associate-/r* 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in a around 0 71.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-74}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-74}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) b)))
   (if (<= a -3.8e-74) (* t_0 (/ 0.5 a)) (* t_0 (/ 0.5 b)))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / b;
	double tmp;
	if (a <= -3.8e-74) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / a) / b;
	double tmp;
	if (a <= -3.8e-74) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / b
	tmp = 0
	if a <= -3.8e-74:
		tmp = t_0 * (0.5 / a)
	else:
		tmp = t_0 * (0.5 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / b)
	tmp = 0.0
	if (a <= -3.8e-74)
		tmp = Float64(t_0 * Float64(0.5 / a));
	else
		tmp = Float64(t_0 * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / b;
	tmp = 0.0;
	if (a <= -3.8e-74)
		tmp = t_0 * (0.5 / a);
	else
		tmp = t_0 * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[a, -3.8e-74], N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7999999999999996e-74

   1. Initial program 79.5%

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

      1. associate-*l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/l* 79.5%

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

      4. metadata-eval 79.5%

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

      5. associate-*l/ 79.5%

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

      6. *-lft-identity 79.5%

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

      7. sub-neg 79.5%

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

      8. distribute-neg-frac 79.5%

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

      9. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

      1. metadata-eval 79.5%

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

      2. div-inv 79.5%

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

      3. *-commutative 79.5%

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

      4. clear-num 78.8%

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

      5. frac-times 79.0%

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

      6. *-un-lft-identity 79.0%

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

      7. frac-add 78.9%

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

      8. associate-/r/ 79.0%

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

      9. *-un-lft-identity 79.0%

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

      10. *-commutative 79.0%

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

      11. neg-mul-1 79.0%

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

      12. sub-neg 79.0%

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

      13. flip-+ 99.1%

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

      14. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/r* 99.1%

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

   8. Simplified 99.1%

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

      1. div-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l/ 99.0%

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

      5. *-commutative 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.1%

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

       2. associate-/r* 99.7%

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

       3. associate-/l* 99.7%

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

       4. associate-*l/ 99.6%

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

       5. *-commutative 99.6%

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

       6. associate-/r* 99.5%

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

   12. Simplified 99.5%

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

   13. Taylor expanded in a around inf 85.7%

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

   if -3.7999999999999996e-74 < a

   1. Initial program 77.8%

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

      1. associate-*l* 77.8%

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

      2. *-rgt-identity 77.8%

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

      3. associate-/l* 77.8%

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

      4. metadata-eval 77.8%

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

      5. associate-*l/ 77.8%

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

      6. *-lft-identity 77.8%

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

      7. sub-neg 77.8%

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

      8. distribute-neg-frac 77.8%

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

      9. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

      1. metadata-eval 77.8%

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

      2. div-inv 77.8%

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

      3. *-commutative 77.8%

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

      4. clear-num 77.8%

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

      5. frac-times 77.8%

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

      6. *-un-lft-identity 77.8%

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

      7. frac-add 77.7%

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

      8. associate-/r/ 77.8%

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

      9. *-un-lft-identity 77.8%

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

      10. *-commutative 77.8%

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

      11. neg-mul-1 77.8%

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

      12. sub-neg 77.8%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l/ 98.9%

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

      5. *-commutative 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. associate-*r/ 98.9%

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

       2. associate-/r* 99.7%

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

       3. associate-/l* 99.6%

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

       4. associate-*l/ 99.7%

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

       5. *-commutative 99.7%

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

       6. associate-/r* 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in a around 0 71.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

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 78.3%

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

   1. associate-*l* 78.3%

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

   2. *-rgt-identity 78.3%

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

   3. associate-/l* 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l/ 78.3%

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

   6. *-lft-identity 78.3%

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

   7. sub-neg 78.3%

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

   8. distribute-neg-frac 78.3%

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

   9. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. *-commutative 78.3%

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

   4. clear-num 78.1%

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

   5. frac-times 78.1%

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

   6. *-un-lft-identity 78.1%

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

   7. frac-add 78.1%

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

   8. associate-/r/ 78.2%

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

   9. *-un-lft-identity 78.2%

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

   10. *-commutative 78.2%

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

   11. neg-mul-1 78.2%

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

   12. sub-neg 78.2%

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

   13. flip-+ 99.0%

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

   14. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-/r* 99.0%

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

8. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. metadata-eval 99.0%

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

   3. *-commutative 99.0%

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

   4. times-frac 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 7: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. associate-*l* 78.3%

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

   2. *-rgt-identity 78.3%

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

   3. associate-/l* 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l/ 78.3%

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

   6. *-lft-identity 78.3%

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

   7. sub-neg 78.3%

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

   8. distribute-neg-frac 78.3%

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

   9. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. *-commutative 78.3%

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

   4. clear-num 78.1%

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

   5. frac-times 78.1%

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

   6. *-un-lft-identity 78.1%

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

   7. frac-add 78.1%

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

   8. associate-/r/ 78.2%

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

   9. *-un-lft-identity 78.2%

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

   10. *-commutative 78.2%

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

   11. neg-mul-1 78.2%

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

   12. sub-neg 78.2%

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

   13. flip-+ 99.0%

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

   14. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-/r* 99.0%

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

8. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. metadata-eval 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-*l/ 98.9%

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

   5. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate-*r/ 99.0%

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

    2. associate-/r* 99.7%

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

    3. associate-/l* 99.6%

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

    4. associate-*l/ 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-/r* 99.6%

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

12. Simplified 99.6%

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

Alternative 8: 99.0% 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 78.3%

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

   1. *-commutative 78.3%

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

   2. associate-*r* 78.3%

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

   3. associate-*r/ 78.3%

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

   4. associate-*r* 78.3%

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

   5. *-rgt-identity 78.3%

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

   6. sub-neg 78.3%

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

   7. distribute-neg-frac 78.3%

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

   8. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. *-commutative 78.3%

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

   2. associate-*r/ 78.3%

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

   3. div-inv 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l* 78.3%

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

   6. *-commutative 78.3%

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 9: 63.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. associate-*l* 78.3%

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

   2. *-rgt-identity 78.3%

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

   3. associate-/l* 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l/ 78.3%

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

   6. *-lft-identity 78.3%

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

   7. sub-neg 78.3%

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

   8. distribute-neg-frac 78.3%

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

   9. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. *-commutative 78.3%

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

   4. clear-num 78.1%

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

   5. frac-times 78.1%

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

   6. *-un-lft-identity 78.1%

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

   7. frac-add 78.1%

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

   8. associate-/r/ 78.2%

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

   9. *-un-lft-identity 78.2%

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

   10. *-commutative 78.2%

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

   11. neg-mul-1 78.2%

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

   12. sub-neg 78.2%

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

   13. flip-+ 99.0%

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

   14. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-/r* 99.0%

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

8. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. metadata-eval 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-*l/ 98.9%

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

   5. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate-*r/ 99.0%

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

    2. associate-/r* 99.7%

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

    3. associate-/l* 99.6%

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

    4. associate-*l/ 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-/r* 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in a around inf 61.7%

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

14. Final simplification 61.7%

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

Reproduce

herbie shell --seed 2024185 
(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))))