NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.7%

Time: 9.3s

Alternatives: 7

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: 78.2% 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: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-*l* 75.0%

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

   2. associate-*l/ 75.1%

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

   3. *-lft-identity 75.1%

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

   4. difference-of-squares 87.6%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. +-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*r/ 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= b 4.8e-99) (* t_0 (/ PI a)) (* (/ PI b) t_0))))

C

double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 4.8e-99) {
		tmp = t_0 * (((double) M_PI) / a);
	} else {
		tmp = (((double) M_PI) / b) * t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 4.8e-99) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = (Math.PI / b) * t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 / (a * b)
	tmp = 0
	if b <= 4.8e-99:
		tmp = t_0 * (math.pi / a)
	else:
		tmp = (math.pi / b) * t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 / Float64(a * b))
	tmp = 0.0
	if (b <= 4.8e-99)
		tmp = Float64(t_0 * Float64(pi / a));
	else
		tmp = Float64(Float64(pi / b) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 / (a * b);
	tmp = 0.0;
	if (b <= 4.8e-99)
		tmp = t_0 * (pi / a);
	else
		tmp = (pi / b) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8e-99], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000001e-99

   1. Initial program 71.8%

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

      1. associate-*l* 71.8%

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

      2. associate-*l/ 71.8%

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

      3. *-lft-identity 71.8%

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

      4. difference-of-squares 85.2%

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

      5. associate-/l/ 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      4. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.9%

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

       2. *-commutative 98.9%

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

       3. times-frac 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around inf 72.2%

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

   if 4.8000000000000001e-99 < b

   1. Initial program 80.9%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-*l/ 80.9%

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

      3. *-lft-identity 80.9%

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

      4. difference-of-squares 91.7%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.8%

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

       2. *-commutative 98.8%

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

       3. times-frac 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around 0 84.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.8e-98) (* (/ PI (* a b)) (/ 0.5 a)) (* (/ PI b) (/ 0.5 (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.8e-98) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.8e-98) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.8e-98:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.8e-98)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.8e-98)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.8e-98], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000001e-98

   1. Initial program 71.8%

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

      1. associate-*l* 71.8%

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

      2. associate-*l/ 71.8%

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

      3. *-lft-identity 71.8%

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

      4. difference-of-squares 85.2%

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

      5. associate-/l/ 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      4. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.9%

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

       2. *-commutative 98.9%

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

       3. *-commutative 98.9%

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

       4. times-frac 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in a around inf 72.3%

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

   if 4.8000000000000001e-98 < b

   1. Initial program 80.9%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-*l/ 80.9%

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

      3. *-lft-identity 80.9%

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

      4. difference-of-squares 91.7%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.8%

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

       2. *-commutative 98.8%

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

       3. times-frac 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around 0 84.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot b}\\ \mathbf{if}\;b \leq 5 \cdot 10^{-98}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (* a b))))
   (if (<= b 5e-98) (* t_0 (/ 0.5 a)) (* t_0 (/ 0.5 b)))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) / (a * b);
	double tmp;
	if (b <= 5e-98) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI / (a * b);
	double tmp;
	if (b <= 5e-98) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi / (a * b)
	tmp = 0
	if b <= 5e-98:
		tmp = t_0 * (0.5 / a)
	else:
		tmp = t_0 * (0.5 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi / Float64(a * b))
	tmp = 0.0
	if (b <= 5e-98)
		tmp = Float64(t_0 * Float64(0.5 / a));
	else
		tmp = Float64(t_0 * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi / (a * b);
	tmp = 0.0;
	if (b <= 5e-98)
		tmp = t_0 * (0.5 / a);
	else
		tmp = t_0 * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 5e-98], N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.00000000000000018e-98

   1. Initial program 71.8%

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

      1. associate-*l* 71.8%

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

      2. associate-*l/ 71.8%

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

      3. *-lft-identity 71.8%

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

      4. difference-of-squares 85.2%

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

      5. associate-/l/ 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      4. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.9%

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

       2. *-commutative 98.9%

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

       3. *-commutative 98.9%

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

       4. times-frac 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in a around inf 72.3%

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

   if 5.00000000000000018e-98 < b

   1. Initial program 80.9%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-*l/ 80.9%

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

      3. *-lft-identity 80.9%

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

      4. difference-of-squares 91.7%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.8%

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

       2. *-commutative 98.8%

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

       3. *-commutative 98.8%

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

       4. times-frac 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around 0 84.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e-98) (* (/ PI (* a b)) (/ 0.5 a)) (* (/ (/ PI b) a) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e-98) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = ((((double) M_PI) / b) / a) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5e-98) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = ((Math.PI / b) / a) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e-98:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = ((math.pi / b) / a) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e-98)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(pi / b) / a) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e-98)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = ((pi / b) / a) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e-98], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.00000000000000018e-98

   1. Initial program 71.8%

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

      1. associate-*l* 71.8%

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

      2. associate-*l/ 71.8%

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

      3. *-lft-identity 71.8%

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

      4. difference-of-squares 85.2%

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

      5. associate-/l/ 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      4. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.9%

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

       2. *-commutative 98.9%

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

       3. *-commutative 98.9%

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

       4. times-frac 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in a around inf 72.3%

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

   if 5.00000000000000018e-98 < b

   1. Initial program 80.9%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-*l/ 80.9%

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

      3. *-lft-identity 80.9%

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

      4. difference-of-squares 91.7%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   9. Simplified 99.7%

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

       1. associate-/l/ 98.8%

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

       2. *-commutative 98.8%

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

       3. associate-*r* 98.8%

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

       4. times-frac 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around 0 84.5%

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

       1. associate-/l/ 84.5%

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

   14. Simplified 84.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-*l* 75.0%

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

   2. associate-*l/ 75.1%

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

   3. *-lft-identity 75.1%

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

   4. difference-of-squares 87.6%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. +-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*r/ 99.7%

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

9. Simplified 99.7%

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

    1. associate-/l/ 98.9%

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

    2. *-commutative 98.9%

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

    3. *-commutative 98.9%

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

    4. times-frac 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 7: 63.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-*l* 75.0%

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

   2. associate-*l/ 75.1%

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

   3. *-lft-identity 75.1%

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

   4. difference-of-squares 87.6%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. +-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*r/ 99.7%

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

9. Simplified 99.7%

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

    1. associate-/l/ 98.9%

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

    2. *-commutative 98.9%

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

    3. times-frac 99.6%

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around inf 60.6%

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

13. Final simplification 60.6%

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

Reproduce

herbie shell --seed 2024020 
(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))))