NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 10.9s

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: 78.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. associate-*l* 82.1%

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

   2. *-rgt-identity 82.1%

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

   3. associate-/l* 82.1%

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

   4. metadata-eval 82.1%

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

   5. associate-*l/ 82.2%

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

   6. *-lft-identity 82.2%

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

   7. sub-neg 82.2%

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

   8. distribute-neg-frac 82.2%

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

   9. metadata-eval 82.2%

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

3. Simplified 82.2%

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

   1. metadata-eval 82.2%

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

   2. div-inv 82.2%

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

   3. *-commutative 82.2%

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

   4. clear-num 82.1%

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

   5. frac-times 82.2%

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

   6. *-un-lft-identity 82.2%

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

   7. frac-add 82.2%

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

   8. associate-/r/ 82.2%

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

   9. *-un-lft-identity 82.2%

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

   10. *-commutative 82.2%

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

   11. neg-mul-1 82.2%

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

   12. sub-neg 82.2%

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

   13. flip-+ 99.3%

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

   14. +-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-/r* 99.3%

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

8. Simplified 99.3%

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

   1. associate-/r* 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. *-commutative 99.6%

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

   5. div-inv 99.7%

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

   6. *-commutative 99.7%

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

   7. metadata-eval 99.7%

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

   8. div-inv 99.7%

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

   9. *-un-lft-identity 99.7%

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

   10. times-frac 99.7%

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

   11. div-inv 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a + b}\\ \mathbf{if}\;b \leq 4.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{t\_0}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{t\_0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (+ a b))))
   (if (<= b 4.3e-40) (* (/ 0.5 a) (/ t_0 b)) (* (/ 0.5 b) (/ t_0 a)))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) / (a + b);
	double tmp;
	if (b <= 4.3e-40) {
		tmp = (0.5 / a) * (t_0 / b);
	} else {
		tmp = (0.5 / b) * (t_0 / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI / (a + b);
	double tmp;
	if (b <= 4.3e-40) {
		tmp = (0.5 / a) * (t_0 / b);
	} else {
		tmp = (0.5 / b) * (t_0 / a);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi / (a + b)
	tmp = 0
	if b <= 4.3e-40:
		tmp = (0.5 / a) * (t_0 / b)
	else:
		tmp = (0.5 / b) * (t_0 / a)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi / Float64(a + b))
	tmp = 0.0
	if (b <= 4.3e-40)
		tmp = Float64(Float64(0.5 / a) * Float64(t_0 / b));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(t_0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi / (a + b);
	tmp = 0.0;
	if (b <= 4.3e-40)
		tmp = (0.5 / a) * (t_0 / b);
	else
		tmp = (0.5 / b) * (t_0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.3e-40], N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.3000000000000003e-40

   1. Initial program 80.7%

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

      1. associate-*l* 80.8%

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

      2. *-rgt-identity 80.8%

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

      3. associate-/l* 80.8%

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

      4. metadata-eval 80.8%

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

      5. associate-*l/ 80.8%

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

      6. *-lft-identity 80.8%

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

      7. sub-neg 80.8%

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

      8. distribute-neg-frac 80.8%

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

      9. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

      1. metadata-eval 80.8%

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

      2. div-inv 80.8%

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

      3. *-commutative 80.8%

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

      4. clear-num 80.8%

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

      5. frac-times 80.8%

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

      6. *-un-lft-identity 80.8%

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

      7. frac-add 80.8%

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

      8. associate-/r/ 80.8%

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

      9. *-un-lft-identity 80.8%

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

      10. *-commutative 80.8%

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

      11. neg-mul-1 80.8%

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

      12. sub-neg 80.8%

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

      13. flip-+ 99.2%

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

      14. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-/r* 99.2%

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

   8. Simplified 99.2%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 99.6%

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

      6. times-frac 96.8%

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

   10. Applied egg-rr 96.8%

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

   if 4.3000000000000003e-40 < b

   1. Initial program 85.3%

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

      1. associate-*l* 85.1%

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

      2. *-rgt-identity 85.1%

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

      3. associate-/l* 85.1%

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

      4. metadata-eval 85.1%

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

      5. associate-*l/ 85.2%

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

      6. *-lft-identity 85.2%

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

      7. sub-neg 85.2%

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

      8. distribute-neg-frac 85.2%

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

      9. metadata-eval 85.2%

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

   3. Simplified 85.2%

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

      1. metadata-eval 85.2%

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

      2. div-inv 85.2%

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

      3. *-commutative 85.2%

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

      4. clear-num 85.1%

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

      5. frac-times 85.2%

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

      6. *-un-lft-identity 85.2%

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

      7. frac-add 85.2%

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

      8. associate-/r/ 85.1%

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

      9. *-un-lft-identity 85.1%

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

      10. *-commutative 85.1%

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

      11. neg-mul-1 85.1%

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

      12. sub-neg 85.1%

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

      13. flip-+ 99.6%

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

      14. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 99.6%

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

      6. *-commutative 99.6%

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

      7. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

Alternative 3: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{a + b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.2e+97)
   (* (/ 0.5 a) (/ (/ PI (+ a b)) b))
   (* (/ 0.5 b) (/ (/ PI a) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.2e+97) {
		tmp = (0.5 / a) * ((((double) M_PI) / (a + b)) / b);
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.2e+97) {
		tmp = (0.5 / a) * ((Math.PI / (a + b)) / b);
	} else {
		tmp = (0.5 / b) * ((Math.PI / a) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.2e+97:
		tmp = (0.5 / a) * ((math.pi / (a + b)) / b)
	else:
		tmp = (0.5 / b) * ((math.pi / a) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.2e+97)
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(pi / Float64(a + b)) / b));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.2e+97)
		tmp = (0.5 / a) * ((pi / (a + b)) / b);
	else
		tmp = (0.5 / b) * ((pi / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.2e+97], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2e97

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. *-rgt-identity 83.7%

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

      3. associate-/l* 83.7%

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

      4. metadata-eval 83.7%

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

      5. associate-*l/ 83.7%

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

      6. *-lft-identity 83.7%

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

      7. sub-neg 83.7%

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

      8. distribute-neg-frac 83.7%

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

      9. metadata-eval 83.7%

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

   3. Simplified 83.7%

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

      1. metadata-eval 83.7%

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

      2. div-inv 83.7%

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

      3. *-commutative 83.7%

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

      4. clear-num 83.7%

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

      5. frac-times 83.7%

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

      6. *-un-lft-identity 83.7%

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

      7. frac-add 83.7%

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

      8. associate-/r/ 83.7%

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

      9. *-un-lft-identity 83.7%

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

      10. *-commutative 83.7%

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

      11. neg-mul-1 83.7%

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

      12. sub-neg 83.7%

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

      13. flip-+ 99.2%

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

      14. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-/r* 99.2%

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

   8. Simplified 99.2%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 99.6%

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

      6. times-frac 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 1.2e97 < b

   1. Initial program 75.3%

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

      1. associate-*l* 75.2%

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

      2. *-rgt-identity 75.2%

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

      3. associate-/l* 75.2%

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

      4. metadata-eval 75.2%

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

      5. associate-*l/ 75.2%

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

      6. *-lft-identity 75.2%

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

      7. sub-neg 75.2%

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

      8. distribute-neg-frac 75.2%

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

      9. metadata-eval 75.2%

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

   3. Simplified 75.2%

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

      1. metadata-eval 75.2%

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

      2. div-inv 75.2%

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

      3. *-commutative 75.2%

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

      4. clear-num 75.2%

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

      5. frac-times 75.3%

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

      6. *-un-lft-identity 75.3%

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

      7. frac-add 75.3%

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

      8. associate-/r/ 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. neg-mul-1 75.2%

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

      12. sub-neg 75.2%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 99.7%

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

       1. div-inv 99.7%

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

       2. metadata-eval 99.7%

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

       3. *-commutative 99.7%

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

       4. times-frac 99.8%

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

       5. associate-/r* 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{a + b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e-73) (/ (/ 0.5 a) (* b (/ a PI))) (* (/ 0.5 b) (/ (/ PI a) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e-73) {
		tmp = (0.5 / a) / (b * (a / ((double) M_PI)));
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9e-73) {
		tmp = (0.5 / a) / (b * (a / Math.PI));
	} else {
		tmp = (0.5 / b) * ((Math.PI / a) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e-73:
		tmp = (0.5 / a) / (b * (a / math.pi))
	else:
		tmp = (0.5 / b) * ((math.pi / a) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e-73)
		tmp = Float64(Float64(0.5 / a) / Float64(b * Float64(a / pi)));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e-73)
		tmp = (0.5 / a) / (b * (a / pi));
	else
		tmp = (0.5 / b) * ((pi / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e-73], N[(N[(0.5 / a), $MachinePrecision] / N[(b * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9e-73

   1. Initial program 81.6%

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

      1. associate-*l* 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. associate-/l* 81.7%

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

      4. metadata-eval 81.7%

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

      5. associate-*l/ 81.8%

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

      6. *-lft-identity 81.8%

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

      7. sub-neg 81.8%

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

      8. distribute-neg-frac 81.8%

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

      9. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

      1. metadata-eval 81.8%

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

      2. div-inv 81.8%

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

      3. *-commutative 81.8%

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

      4. clear-num 81.7%

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

      5. frac-times 81.7%

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

      6. *-un-lft-identity 81.7%

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

      7. frac-add 81.7%

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

      8. associate-/r/ 81.7%

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

      9. *-un-lft-identity 81.7%

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

      10. *-commutative 81.7%

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

      11. neg-mul-1 81.7%

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

      12. sub-neg 81.7%

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

      13. flip-+ 98.3%

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

      14. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-/r* 98.3%

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

   8. Simplified 98.3%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l* 99.5%

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

      6. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around inf 85.9%

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

       1. associate-/r* 85.8%

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

   13. Simplified 85.8%

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

       1. clear-num 85.8%

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

       2. un-div-inv 85.9%

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

       3. div-inv 85.8%

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

       4. clear-num 85.9%

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

   15. Applied egg-rr 85.9%

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

   if -9e-73 < a

   1. Initial program 82.4%

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

      1. associate-*l* 82.3%

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

      2. *-rgt-identity 82.3%

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

      3. associate-/l* 82.3%

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

      4. metadata-eval 82.3%

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

      5. associate-*l/ 82.4%

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

      6. *-lft-identity 82.4%

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

      7. sub-neg 82.4%

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

      8. distribute-neg-frac 82.4%

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

      9. metadata-eval 82.4%

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

   3. Simplified 82.4%

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

      1. metadata-eval 82.4%

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

      2. div-inv 82.4%

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

      3. *-commutative 82.4%

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

      4. clear-num 82.3%

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

      5. frac-times 82.4%

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

      6. *-un-lft-identity 82.4%

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

      7. frac-add 82.4%

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

      8. associate-/r/ 82.4%

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

      9. *-un-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. neg-mul-1 82.4%

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

      12. sub-neg 82.4%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 73.2%

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

       1. div-inv 73.2%

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

       2. metadata-eval 73.2%

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

       3. *-commutative 73.2%

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

       4. times-frac 73.0%

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

       5. associate-/r* 73.0%

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

   11. Applied egg-rr 73.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b}\\ \mathbf{if}\;a \leq -7 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5}{a} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) b)))
   (if (<= a -7e-73) (* (/ 0.5 a) t_0) (* (/ 0.5 b) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / b;
	double tmp;
	if (a <= -7e-73) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = (0.5 / b) * t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / a) / b;
	double tmp;
	if (a <= -7e-73) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = (0.5 / b) * t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / b
	tmp = 0
	if a <= -7e-73:
		tmp = (0.5 / a) * t_0
	else:
		tmp = (0.5 / b) * t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / b)
	tmp = 0.0
	if (a <= -7e-73)
		tmp = Float64(Float64(0.5 / a) * t_0);
	else
		tmp = Float64(Float64(0.5 / b) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / b;
	tmp = 0.0;
	if (a <= -7e-73)
		tmp = (0.5 / a) * t_0;
	else
		tmp = (0.5 / b) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[a, -7e-73], N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.9999999999999995e-73

   1. Initial program 81.6%

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

      1. associate-*l* 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. associate-/l* 81.7%

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

      4. metadata-eval 81.7%

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

      5. associate-*l/ 81.8%

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

      6. *-lft-identity 81.8%

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

      7. sub-neg 81.8%

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

      8. distribute-neg-frac 81.8%

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

      9. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

      1. metadata-eval 81.8%

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

      2. div-inv 81.8%

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

      3. *-commutative 81.8%

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

      4. clear-num 81.7%

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

      5. frac-times 81.7%

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

      6. *-un-lft-identity 81.7%

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

      7. frac-add 81.7%

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

      8. associate-/r/ 81.7%

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

      9. *-un-lft-identity 81.7%

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

      10. *-commutative 81.7%

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

      11. neg-mul-1 81.7%

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

      12. sub-neg 81.7%

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

      13. flip-+ 98.3%

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

      14. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-/r* 98.3%

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

   8. Simplified 98.3%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l* 99.5%

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

      6. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around inf 85.9%

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

       1. associate-/r* 85.8%

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

   13. Simplified 85.8%

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

   if -6.9999999999999995e-73 < a

   1. Initial program 82.4%

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

      1. associate-*l* 82.3%

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

      2. *-rgt-identity 82.3%

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

      3. associate-/l* 82.3%

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

      4. metadata-eval 82.3%

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

      5. associate-*l/ 82.4%

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

      6. *-lft-identity 82.4%

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

      7. sub-neg 82.4%

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

      8. distribute-neg-frac 82.4%

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

      9. metadata-eval 82.4%

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

   3. Simplified 82.4%

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

      1. metadata-eval 82.4%

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

      2. div-inv 82.4%

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

      3. *-commutative 82.4%

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

      4. clear-num 82.3%

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

      5. frac-times 82.4%

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

      6. *-un-lft-identity 82.4%

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

      7. frac-add 82.4%

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

      8. associate-/r/ 82.4%

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

      9. *-un-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. neg-mul-1 82.4%

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

      12. sub-neg 82.4%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 73.2%

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

       1. div-inv 73.2%

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

       2. metadata-eval 73.2%

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

       3. *-commutative 73.2%

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

       4. times-frac 73.0%

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

       5. associate-/r* 73.0%

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

   11. Applied egg-rr 73.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e-73) (* (/ 0.5 a) (/ (/ PI a) b)) (* PI (/ 0.5 (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e-73) {
		tmp = (0.5 / a) * ((((double) M_PI) / 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 <= -9e-73) {
		tmp = (0.5 / a) * ((Math.PI / a) / b);
	} else {
		tmp = Math.PI * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e-73:
		tmp = (0.5 / a) * ((math.pi / a) / b)
	else:
		tmp = math.pi * (0.5 / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e-73)
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(pi / a) / b));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e-73)
		tmp = (0.5 / a) * ((pi / a) / b);
	else
		tmp = pi * (0.5 / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e-73], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9e-73

   1. Initial program 81.6%

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

      1. associate-*l* 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. associate-/l* 81.7%

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

      4. metadata-eval 81.7%

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

      5. associate-*l/ 81.8%

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

      6. *-lft-identity 81.8%

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

      7. sub-neg 81.8%

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

      8. distribute-neg-frac 81.8%

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

      9. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

      1. metadata-eval 81.8%

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

      2. div-inv 81.8%

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

      3. *-commutative 81.8%

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

      4. clear-num 81.7%

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

      5. frac-times 81.7%

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

      6. *-un-lft-identity 81.7%

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

      7. frac-add 81.7%

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

      8. associate-/r/ 81.7%

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

      9. *-un-lft-identity 81.7%

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

      10. *-commutative 81.7%

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

      11. neg-mul-1 81.7%

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

      12. sub-neg 81.7%

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

      13. flip-+ 98.3%

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

      14. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-/r* 98.3%

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

   8. Simplified 98.3%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l* 99.5%

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

      6. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around inf 85.9%

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

       1. associate-/r* 85.8%

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

   13. Simplified 85.8%

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

   if -9e-73 < a

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*r* 82.3%

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

      3. associate-*r/ 82.4%

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

      4. associate-*r* 82.4%

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

      5. *-rgt-identity 82.4%

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

      6. sub-neg 82.4%

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

      7. distribute-neg-frac 82.4%

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

      8. metadata-eval 82.4%

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

   3. Simplified 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*r/ 82.4%

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

      3. div-inv 82.4%

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

      4. metadata-eval 82.4%

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

      5. associate-*l* 82.4%

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

      6. *-commutative 82.4%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 73.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e-73) (* (/ 0.5 a) (/ (/ PI a) b)) (* (/ 0.5 a) (/ (/ PI b) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e-73) {
		tmp = (0.5 / a) * ((((double) M_PI) / a) / b);
	} else {
		tmp = (0.5 / a) * ((((double) M_PI) / b) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9e-73) {
		tmp = (0.5 / a) * ((Math.PI / a) / b);
	} else {
		tmp = (0.5 / a) * ((Math.PI / b) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e-73:
		tmp = (0.5 / a) * ((math.pi / a) / b)
	else:
		tmp = (0.5 / a) * ((math.pi / b) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e-73)
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(pi / a) / b));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(pi / b) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e-73)
		tmp = (0.5 / a) * ((pi / a) / b);
	else
		tmp = (0.5 / a) * ((pi / b) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e-73], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9e-73

   1. Initial program 81.6%

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

      1. associate-*l* 81.7%

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

      2. *-rgt-identity 81.7%

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

      3. associate-/l* 81.7%

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

      4. metadata-eval 81.7%

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

      5. associate-*l/ 81.8%

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

      6. *-lft-identity 81.8%

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

      7. sub-neg 81.8%

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

      8. distribute-neg-frac 81.8%

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

      9. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

      1. metadata-eval 81.8%

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

      2. div-inv 81.8%

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

      3. *-commutative 81.8%

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

      4. clear-num 81.7%

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

      5. frac-times 81.7%

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

      6. *-un-lft-identity 81.7%

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

      7. frac-add 81.7%

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

      8. associate-/r/ 81.7%

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

      9. *-un-lft-identity 81.7%

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

      10. *-commutative 81.7%

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

      11. neg-mul-1 81.7%

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

      12. sub-neg 81.7%

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

      13. flip-+ 98.3%

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

      14. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-/r* 98.3%

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

   8. Simplified 98.3%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l* 99.5%

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

      6. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around inf 85.9%

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

       1. associate-/r* 85.8%

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

   13. Simplified 85.8%

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

   if -9e-73 < a

   1. Initial program 82.4%

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

      1. associate-*l* 82.3%

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

      2. *-rgt-identity 82.3%

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

      3. associate-/l* 82.3%

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

      4. metadata-eval 82.3%

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

      5. associate-*l/ 82.4%

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

      6. *-lft-identity 82.4%

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

      7. sub-neg 82.4%

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

      8. distribute-neg-frac 82.4%

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

      9. metadata-eval 82.4%

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

   3. Simplified 82.4%

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

      1. metadata-eval 82.4%

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

      2. div-inv 82.4%

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

      3. *-commutative 82.4%

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

      4. clear-num 82.3%

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

      5. frac-times 82.4%

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

      6. *-un-lft-identity 82.4%

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

      7. frac-add 82.4%

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

      8. associate-/r/ 82.4%

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

      9. *-un-lft-identity 82.4%

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

      10. *-commutative 82.4%

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

      11. neg-mul-1 82.4%

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

      12. sub-neg 82.4%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 99.6%

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

      6. times-frac 93.2%

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

   10. Applied egg-rr 93.2%

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

   11. Taylor expanded in a around 0 66.4%

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

Alternative 8: 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 82.2%

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

   1. associate-*l* 82.1%

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

   2. *-rgt-identity 82.1%

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

   3. associate-/l* 82.1%

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

   4. metadata-eval 82.1%

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

   5. associate-*l/ 82.2%

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

   6. *-lft-identity 82.2%

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

   7. sub-neg 82.2%

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

   8. distribute-neg-frac 82.2%

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

   9. metadata-eval 82.2%

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

3. Simplified 82.2%

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

   1. metadata-eval 82.2%

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

   2. div-inv 82.2%

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

   3. *-commutative 82.2%

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

   4. clear-num 82.1%

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

   5. frac-times 82.2%

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

   6. *-un-lft-identity 82.2%

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

   7. frac-add 82.2%

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

   8. associate-/r/ 82.2%

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

   9. *-un-lft-identity 82.2%

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

   10. *-commutative 82.2%

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

   11. neg-mul-1 82.2%

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

   12. sub-neg 82.2%

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

   13. flip-+ 99.3%

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

   14. +-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-/r* 99.3%

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

8. Simplified 99.3%

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

   1. div-inv 99.3%

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

   2. metadata-eval 99.3%

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

   3. *-commutative 99.3%

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

   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 9: 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 82.2%

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

   1. *-commutative 82.2%

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

   2. associate-*r* 82.2%

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

   3. associate-*r/ 82.2%

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

   4. associate-*r* 82.2%

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

   5. *-rgt-identity 82.2%

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

   6. sub-neg 82.2%

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

   7. distribute-neg-frac 82.2%

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

   8. metadata-eval 82.2%

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

3. Simplified 82.2%

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

   1. *-commutative 82.2%

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

   2. associate-*r/ 82.2%

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

   3. div-inv 82.2%

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

   4. metadata-eval 82.2%

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

   5. associate-*l* 82.2%

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

   6. *-commutative 82.2%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 10: 63.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. associate-*l* 82.1%

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

   2. *-rgt-identity 82.1%

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

   3. associate-/l* 82.1%

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

   4. metadata-eval 82.1%

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

   5. associate-*l/ 82.2%

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

   6. *-lft-identity 82.2%

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

   7. sub-neg 82.2%

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

   8. distribute-neg-frac 82.2%

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

   9. metadata-eval 82.2%

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

3. Simplified 82.2%

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

   1. metadata-eval 82.2%

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

   2. div-inv 82.2%

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

   3. *-commutative 82.2%

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

   4. clear-num 82.1%

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

   5. frac-times 82.2%

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

   6. *-un-lft-identity 82.2%

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

   7. frac-add 82.2%

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

   8. associate-/r/ 82.2%

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

   9. *-un-lft-identity 82.2%

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

   10. *-commutative 82.2%

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

   11. neg-mul-1 82.2%

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

   12. sub-neg 82.2%

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

   13. flip-+ 99.3%

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

   14. +-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-/r* 99.3%

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

8. Simplified 99.3%

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

   1. associate-/r* 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-/l* 99.6%

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

   6. times-frac 95.0%

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

10. Applied egg-rr 95.0%

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

11. Taylor expanded in a around inf 61.9%

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

    1. associate-/r* 61.9%

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

13. Simplified 61.9%

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

Reproduce

herbie shell --seed 2024182 
(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))))