NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 13.1s

Alternatives: 11

Speedup: 1.6×

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: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. un-div-inv 82.0%

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

   2. difference-of-squares 90.2%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

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

4. Applied egg-rr 90.2%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

8. Applied egg-rr 99.6%

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

Alternative 2: 78.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.15 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(b - a\right)} \cdot \frac{0.5}{b + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.15e-142)
   (/ (/ (* PI -0.5) (* b a)) (- b a))
   (* (/ PI (* a (- b a))) (/ 0.5 (+ b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.15e-142) {
		tmp = ((((double) M_PI) * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (((double) M_PI) / (a * (b - a))) * (0.5 / (b + a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.15e-142) {
		tmp = ((Math.PI * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (Math.PI / (a * (b - a))) * (0.5 / (b + a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.15e-142:
		tmp = ((math.pi * -0.5) / (b * a)) / (b - a)
	else:
		tmp = (math.pi / (a * (b - a))) * (0.5 / (b + a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.15e-142)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(pi / Float64(a * Float64(b - a))) * Float64(0.5 / Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.15e-142)
		tmp = ((pi * -0.5) / (b * a)) / (b - a);
	else
		tmp = (pi / (a * (b - a))) * (0.5 / (b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.15e-142], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1499999999999999e-142

   1. Initial program 86.0%

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

      1. un-div-inv 86.0%

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

      2. difference-of-squares 90.4%

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

      3. associate-/r* 90.5%

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

      4. div-inv 90.5%

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

      5. metadata-eval 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around 0 83.6%

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

      1. associate-*r/ 83.6%

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

   9. Simplified 83.6%

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

   if -3.1499999999999999e-142 < a

   1. Initial program 79.6%

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

      1. un-div-inv 79.7%

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

      2. difference-of-squares 90.0%

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

      3. associate-/r* 90.0%

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

      4. div-inv 90.0%

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

      5. metadata-eval 90.0%

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

   4. Applied egg-rr 90.0%

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

   5. Taylor expanded in a around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-/l/ 61.7%

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

      3. frac-times 61.7%

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

      4. *-un-lft-identity 61.7%

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

   7. Applied egg-rr 61.7%

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

      1. associate-*r* 70.2%

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

      2. times-frac 70.1%

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

      3. +-commutative 70.1%

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

   9. Applied egg-rr 70.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.15 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(b - a\right)} \cdot \frac{0.5}{b + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.03 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.03e-112)
   (/ (/ (* PI -0.5) (* b a)) (- b a))
   (/ (/ (* PI 0.5) (* b a)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.03e-112) {
		tmp = ((((double) M_PI) * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = ((((double) M_PI) * 0.5) / (b * a)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.03e-112) {
		tmp = ((Math.PI * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = ((Math.PI * 0.5) / (b * a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.03e-112:
		tmp = ((math.pi * -0.5) / (b * a)) / (b - a)
	else:
		tmp = ((math.pi * 0.5) / (b * a)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.03e-112)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.03e-112)
		tmp = ((pi * -0.5) / (b * a)) / (b - a);
	else
		tmp = ((pi * 0.5) / (b * a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.03e-112], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.03000000000000002e-112

   1. Initial program 82.1%

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

      1. un-div-inv 82.2%

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

      2. difference-of-squares 90.6%

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

      3. associate-/r* 90.7%

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

      4. div-inv 90.7%

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

      5. metadata-eval 90.7%

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

   4. Applied egg-rr 90.7%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around 0 73.8%

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

      1. associate-*r/ 73.8%

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

   9. Simplified 73.8%

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

   if 1.03000000000000002e-112 < b

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. *-rgt-identity 81.4%

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

      3. associate-/l* 81.4%

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

      4. metadata-eval 81.4%

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

      5. associate-*l/ 81.5%

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

      6. *-lft-identity 81.5%

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

      7. sub-neg 81.5%

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

      8. distribute-neg-frac 81.5%

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

      9. metadata-eval 81.5%

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

   3. Simplified 81.5%

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

      1. metadata-eval 81.5%

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

      2. div-inv 81.5%

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

      3. associate-*r/ 81.4%

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

      4. *-commutative 81.4%

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

      5. difference-of-squares 89.0%

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

      6. associate-/r* 99.5%

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

   6. Applied egg-rr 78.2%

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

   7. Taylor expanded in a around 0 78.3%

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

      1. associate-*r/ 78.3%

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

   9. Simplified 78.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.03 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.35 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.35e-113)
   (/ (/ (* PI -0.5) (* b a)) (- b a))
   (/ (* 0.5 (/ PI b)) (* a (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.35e-113) {
		tmp = ((((double) M_PI) * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) / (a * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.35e-113) {
		tmp = ((Math.PI * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (0.5 * (Math.PI / b)) / (a * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.35e-113:
		tmp = ((math.pi * -0.5) / (b * a)) / (b - a)
	else:
		tmp = (0.5 * (math.pi / b)) / (a * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.35e-113)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.35e-113)
		tmp = ((pi * -0.5) / (b * a)) / (b - a);
	else
		tmp = (0.5 * (pi / b)) / (a * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.35e-113], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.34999999999999977e-113

   1. Initial program 82.1%

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

      1. un-div-inv 82.2%

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

      2. difference-of-squares 90.6%

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

      3. associate-/r* 90.7%

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

      4. div-inv 90.7%

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

      5. metadata-eval 90.7%

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

   4. Applied egg-rr 90.7%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around 0 73.8%

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

      1. associate-*r/ 73.8%

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

   9. Simplified 73.8%

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

   if 4.34999999999999977e-113 < b

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. *-rgt-identity 81.4%

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

      3. associate-/l* 81.4%

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

      4. metadata-eval 81.4%

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

      5. associate-*l/ 81.5%

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

      6. *-lft-identity 81.5%

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

      7. sub-neg 81.5%

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

      8. distribute-neg-frac 81.5%

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

      9. metadata-eval 81.5%

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

   3. Simplified 81.5%

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

      1. metadata-eval 81.5%

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

      2. div-inv 81.5%

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

      3. associate-*r/ 81.4%

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

      4. *-commutative 81.4%

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

      5. difference-of-squares 89.0%

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

      6. associate-/r* 99.5%

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

   6. Applied egg-rr 78.2%

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

   7. Taylor expanded in a around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. *-commutative 78.3%

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

      3. times-frac 78.3%

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

   9. Simplified 78.3%

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

       1. *-un-lft-identity 78.3%

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

       2. associate-/l* 68.9%

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

   11. Applied egg-rr 68.9%

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

       1. *-lft-identity 68.9%

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

   13. Simplified 68.9%

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

       1. *-commutative 68.9%

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

       2. frac-times 78.2%

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

       3. *-commutative 78.2%

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

       4. associate-*r/ 78.3%

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

       5. *-commutative 78.3%

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

       6. *-un-lft-identity 78.3%

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

       7. times-frac 78.3%

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

       8. metadata-eval 78.3%

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

   15. Applied egg-rr 78.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.35 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-111}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{-0.5}{b}}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e-111)
   (/ (* PI (/ (/ -0.5 b) a)) (- b a))
   (/ (* 0.5 (/ PI b)) (* a (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-111) {
		tmp = (((double) M_PI) * ((-0.5 / b) / a)) / (b - a);
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) / (a * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-111) {
		tmp = (Math.PI * ((-0.5 / b) / a)) / (b - a);
	} else {
		tmp = (0.5 * (Math.PI / b)) / (a * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.15e-111:
		tmp = (math.pi * ((-0.5 / b) / a)) / (b - a)
	else:
		tmp = (0.5 * (math.pi / b)) / (a * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15e-111)
		tmp = Float64(Float64(pi * Float64(Float64(-0.5 / b) / a)) / Float64(b - a));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e-111)
		tmp = (pi * ((-0.5 / b) / a)) / (b - a);
	else
		tmp = (0.5 * (pi / b)) / (a * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15e-111], N[(N[(Pi * N[(N[(-0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e-111

   1. Initial program 82.1%

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

      1. un-div-inv 82.2%

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

      2. difference-of-squares 90.6%

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

      3. associate-/r* 90.7%

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

      4. div-inv 90.7%

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

      5. metadata-eval 90.7%

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

   4. Applied egg-rr 90.7%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around inf 79.2%

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

   8. Taylor expanded in b around 0 73.8%

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

      1. metadata-eval 73.8%

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

      2. distribute-lft-neg-in 73.8%

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

      3. associate-*r/ 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-/l* 73.8%

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

      6. associate-/l/ 73.8%

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

      7. distribute-rgt-neg-in 73.8%

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

      8. distribute-neg-frac 73.8%

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

      9. distribute-neg-frac 73.8%

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

      10. metadata-eval 73.8%

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

   10. Simplified 73.8%

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

   if 1.15e-111 < b

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. *-rgt-identity 81.4%

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

      3. associate-/l* 81.4%

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

      4. metadata-eval 81.4%

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

      5. associate-*l/ 81.5%

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

      6. *-lft-identity 81.5%

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

      7. sub-neg 81.5%

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

      8. distribute-neg-frac 81.5%

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

      9. metadata-eval 81.5%

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

   3. Simplified 81.5%

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

      1. metadata-eval 81.5%

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

      2. div-inv 81.5%

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

      3. associate-*r/ 81.4%

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

      4. *-commutative 81.4%

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

      5. difference-of-squares 89.0%

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

      6. associate-/r* 99.5%

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

   6. Applied egg-rr 78.2%

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

   7. Taylor expanded in a around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. *-commutative 78.3%

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

      3. times-frac 78.3%

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

   9. Simplified 78.3%

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

       1. *-un-lft-identity 78.3%

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

       2. associate-/l* 68.9%

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

   11. Applied egg-rr 68.9%

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

       1. *-lft-identity 68.9%

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

   13. Simplified 68.9%

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

       1. *-commutative 68.9%

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

       2. frac-times 78.2%

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

       3. *-commutative 78.2%

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

       4. associate-*r/ 78.3%

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

       5. *-commutative 78.3%

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

       6. *-un-lft-identity 78.3%

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

       7. times-frac 78.3%

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

       8. metadata-eval 78.3%

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

   15. Applied egg-rr 78.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-111}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{-0.5}{b}}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. un-div-inv 82.0%

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

   2. difference-of-squares 90.2%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

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

4. Applied egg-rr 90.2%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in a around 0 99.6%

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

    1. associate-/r* 99.6%

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

11. Simplified 99.6%

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

Alternative 7: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. un-div-inv 82.0%

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

   2. difference-of-squares 90.2%

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

   3. associate-/r* 90.2%

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

   4. div-inv 90.2%

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

   5. metadata-eval 90.2%

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

4. Applied egg-rr 90.2%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in a around 0 99.6%

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

10. Final simplification 99.6%

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

Alternative 8: 67.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. *-rgt-identity 81.9%

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

   3. associate-/l* 81.9%

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

   4. metadata-eval 81.9%

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

   5. associate-*l/ 82.0%

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

   6. *-lft-identity 82.0%

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

   7. sub-neg 82.0%

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

   8. distribute-neg-frac 82.0%

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

   9. metadata-eval 82.0%

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

3. Simplified 82.0%

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

   1. metadata-eval 82.0%

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

   2. div-inv 82.0%

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

   3. associate-*r/ 81.9%

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

   4. *-commutative 81.9%

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

   5. difference-of-squares 90.1%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 61.0%

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

7. Taylor expanded in a around 0 61.1%

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

   1. associate-*r/ 61.1%

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

   2. *-commutative 61.1%

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

   3. times-frac 61.0%

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

9. Simplified 61.0%

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

    1. *-un-lft-identity 61.0%

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

    2. associate-/l* 55.6%

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

11. Applied egg-rr 55.6%

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

    1. *-lft-identity 55.6%

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

13. Simplified 55.6%

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

    1. *-commutative 55.6%

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

    2. frac-times 61.1%

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

    3. *-commutative 61.1%

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

    4. associate-*r/ 61.1%

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

    5. *-commutative 61.1%

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

    6. *-un-lft-identity 61.1%

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

    7. times-frac 61.1%

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

    8. metadata-eval 61.1%

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

15. Applied egg-rr 61.1%

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

16. Final simplification 61.1%

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

Alternative 9: 66.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. *-rgt-identity 81.9%

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

   3. associate-/l* 81.9%

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

   4. metadata-eval 81.9%

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

   5. associate-*l/ 82.0%

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

   6. *-lft-identity 82.0%

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

   7. sub-neg 82.0%

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

   8. distribute-neg-frac 82.0%

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

   9. metadata-eval 82.0%

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

3. Simplified 82.0%

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

   1. metadata-eval 82.0%

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

   2. div-inv 82.0%

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

   3. associate-*r/ 81.9%

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

   4. *-commutative 81.9%

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

   5. difference-of-squares 90.1%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 61.0%

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

7. Taylor expanded in a around 0 61.1%

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

   1. associate-*r/ 61.1%

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

   2. *-commutative 61.1%

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

   3. times-frac 61.0%

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

9. Simplified 61.0%

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

    1. associate-*l/ 61.1%

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

11. Applied egg-rr 61.1%

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

    1. associate-/l/ 61.1%

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

    2. *-commutative 61.1%

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

    3. frac-times 55.6%

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

    4. *-commutative 55.6%

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

    5. frac-times 61.1%

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

    6. *-commutative 61.1%

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

    7. times-frac 61.1%

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

13. Applied egg-rr 61.1%

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

Alternative 10: 61.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. *-rgt-identity 81.9%

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

   3. associate-/l* 81.9%

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

   4. metadata-eval 81.9%

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

   5. associate-*l/ 82.0%

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

   6. *-lft-identity 82.0%

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

   7. sub-neg 82.0%

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

   8. distribute-neg-frac 82.0%

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

   9. metadata-eval 82.0%

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

3. Simplified 82.0%

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

   1. metadata-eval 82.0%

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

   2. div-inv 82.0%

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

   3. associate-*r/ 81.9%

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

   4. *-commutative 81.9%

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

   5. difference-of-squares 90.1%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 61.0%

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

7. Taylor expanded in a around 0 61.1%

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

   1. associate-*r/ 61.1%

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

   2. *-commutative 61.1%

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

   3. times-frac 61.0%

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

9. Simplified 61.0%

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

    1. *-un-lft-identity 61.0%

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

    2. associate-/l* 55.6%

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

11. Applied egg-rr 55.6%

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

    1. *-lft-identity 55.6%

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

13. Simplified 55.6%

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

Alternative 11: 30.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* 81.9%

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

   2. *-rgt-identity 81.9%

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

   3. associate-/l* 81.9%

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

   4. metadata-eval 81.9%

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

   5. associate-*l/ 82.0%

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

   6. *-lft-identity 82.0%

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

   7. sub-neg 82.0%

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

   8. distribute-neg-frac 82.0%

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

   9. metadata-eval 82.0%

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

3. Simplified 82.0%

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

   1. metadata-eval 82.0%

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

   2. div-inv 82.0%

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

   3. associate-*r/ 81.9%

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

   4. *-commutative 81.9%

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

   5. difference-of-squares 90.1%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 61.0%

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

7. Taylor expanded in a around 0 61.1%

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

   1. associate-*r/ 61.1%

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

   2. *-commutative 61.1%

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

   3. times-frac 61.0%

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

9. Simplified 61.0%

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

    1. *-un-lft-identity 61.0%

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

    2. associate-/l* 55.6%

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

11. Applied egg-rr 55.6%

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

    1. *-lft-identity 55.6%

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

13. Simplified 55.6%

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

14. Taylor expanded in b around 0 28.9%

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

15. Final simplification 28.9%

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

Reproduce

herbie shell --seed 2024101 
(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))))