NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 11.2s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi}{a + b}}{2} \cdot \frac{1}{a \cdot b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. inv-pow 78.4%

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

   2. difference-of-squares 87.4%

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

   3. unpow-prod-down 88.1%

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

   4. inv-pow 88.1%

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

   5. inv-pow 88.1%

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

3. Applied egg-rr 88.1%

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

   1. associate-*r/ 88.2%

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

   2. *-rgt-identity 88.2%

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

   3. +-commutative 88.2%

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

5. Simplified 88.2%

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

   1. pow1 88.2%

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

   2. frac-times 88.2%

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

   3. +-commutative 88.2%

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

   4. div-inv 88.3%

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

   5. +-commutative 88.3%

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

   6. inv-pow 88.3%

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

   7. inv-pow 88.3%

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

7. Applied egg-rr 88.3%

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

   1. unpow1 88.3%

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

   2. associate-*l/ 99.6%

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

   3. unpow-1 99.6%

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

   4. unpow-1 99.6%

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

9. Simplified 99.6%

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

    1. times-frac 99.6%

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around 0 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(2 \cdot \left(b - a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.25e-67)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (/ (/ PI b) (* a (* 2.0 (- b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.25e-67) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (((double) M_PI) / b) / (a * (2.0 * (b - a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.25e-67) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (Math.PI / b) / (a * (2.0 * (b - a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.25e-67:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (math.pi / b) / (a * (2.0 * (b - a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.25e-67)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(pi / b) / Float64(a * Float64(2.0 * Float64(b - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.25e-67)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (pi / b) / (a * (2.0 * (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.25e-67], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(a * N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.25000000000000008e-67

   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. Step-by-step derivation

      1. inv-pow 79.6%

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

      2. difference-of-squares 87.0%

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

      3. unpow-prod-down 88.1%

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

      4. inv-pow 88.1%

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

      5. inv-pow 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. associate-*r/ 88.2%

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

      2. *-rgt-identity 88.2%

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

      3. +-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. pow1 88.2%

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

      2. frac-times 88.2%

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

      3. +-commutative 88.2%

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

      4. div-inv 88.2%

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

      5. +-commutative 88.2%

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

      6. inv-pow 88.2%

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

      7. inv-pow 88.2%

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

   7. Applied egg-rr 88.2%

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

      1. unpow1 88.2%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around inf 59.4%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   11. Step-by-step derivation

       1. associate-*r/ 59.4%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

       2. *-commutative 59.4%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

       3. times-frac 59.2%

          \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

       4. unpow2 59.2%

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

   12. Simplified 59.2%

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

       1. associate-*l/ 59.2%

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

   14. Applied egg-rr 59.2%

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

       1. times-frac 69.1%

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

   16. Simplified 69.1%

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

   if 2.25000000000000008e-67 < b

   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. inv-pow 75.1%

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

      2. difference-of-squares 88.3%

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

      3. unpow-prod-down 88.1%

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

      4. inv-pow 88.1%

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

      5. inv-pow 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. associate-*r/ 88.2%

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

      2. *-rgt-identity 88.2%

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

      3. +-commutative 88.2%

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

   5. Simplified 88.2%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. un-div-inv 82.7%

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

      2. frac-times 82.8%

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

      3. div-inv 82.8%

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

   8. Applied egg-rr 82.8%

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

      1. associate-/l/ 94.2%

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

   10. Simplified 94.2%

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

   11. Taylor expanded in a around 0 93.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(2 \cdot \left(b - a\right)\right)}\\ \end{array} \]

Alternative 3: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e-24) (* (/ PI a) (/ (/ 0.5 b) a)) (* (/ 0.5 a) (/ PI (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-24) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / a) * (((double) M_PI) / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-24) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / a) * (Math.PI / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.9e-24:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (0.5 / a) * (math.pi / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.9e-24)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.9e-24)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (0.5 / a) * (pi / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.9e-24], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8999999999999999e-24

   1. Initial program 79.9%

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

      1. inv-pow 79.9%

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

      2. difference-of-squares 87.2%

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

      3. unpow-prod-down 88.3%

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

      4. inv-pow 88.3%

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

      5. inv-pow 88.3%

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

   3. Applied egg-rr 88.3%

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

      1. associate-*r/ 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. +-commutative 88.4%

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

   5. Simplified 88.4%

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

      1. pow1 88.4%

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

      2. frac-times 88.4%

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

      3. +-commutative 88.4%

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

      4. div-inv 88.4%

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

      5. +-commutative 88.4%

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

      6. inv-pow 88.4%

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

      7. inv-pow 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. unpow1 88.4%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around inf 59.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   11. Step-by-step derivation

       1. associate-*r/ 59.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

       2. *-commutative 59.5%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

       3. times-frac 59.3%

          \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

       4. unpow2 59.3%

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

   12. Simplified 59.3%

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

       1. associate-*l/ 59.3%

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

   14. Applied egg-rr 59.3%

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

       1. times-frac 69.1%

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

   16. Simplified 69.1%

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

   if 2.8999999999999999e-24 < b

   1. Initial program 74.0%

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

      1. inv-pow 74.0%

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

      2. difference-of-squares 87.8%

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

      3. unpow-prod-down 87.6%

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

      4. inv-pow 87.6%

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

      5. inv-pow 87.6%

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

   3. Applied egg-rr 87.6%

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

      1. associate-*r/ 87.7%

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

      2. *-rgt-identity 87.7%

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

      3. +-commutative 87.7%

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

   5. Simplified 87.7%

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

   6. Taylor expanded in a around 0 80.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 80.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. times-frac 80.8%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{\pi}{{b}^{2}}} \]

      3. unpow2 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-24) (* (/ PI a) (/ (/ 0.5 b) a)) (* (/ 0.5 b) (/ PI (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-24) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / b) * (((double) M_PI) / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-24) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / b) * (Math.PI / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.5e-24:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (0.5 / b) * (math.pi / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e-24)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(pi / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.5e-24)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (0.5 / b) * (pi / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e-24], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.4999999999999996e-24

   1. Initial program 79.9%

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

      1. inv-pow 79.9%

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

      2. difference-of-squares 87.2%

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

      3. unpow-prod-down 88.3%

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

      4. inv-pow 88.3%

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

      5. inv-pow 88.3%

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

   3. Applied egg-rr 88.3%

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

      1. associate-*r/ 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. +-commutative 88.4%

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

   5. Simplified 88.4%

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

      1. pow1 88.4%

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

      2. frac-times 88.4%

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

      3. +-commutative 88.4%

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

      4. div-inv 88.4%

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

      5. +-commutative 88.4%

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

      6. inv-pow 88.4%

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

      7. inv-pow 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. unpow1 88.4%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around inf 59.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   11. Step-by-step derivation

       1. associate-*r/ 59.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

       2. *-commutative 59.5%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

       3. times-frac 59.3%

          \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

       4. unpow2 59.3%

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

   12. Simplified 59.3%

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

       1. associate-*l/ 59.3%

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

   14. Applied egg-rr 59.3%

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

       1. times-frac 69.1%

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

   16. Simplified 69.1%

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

   if 3.4999999999999996e-24 < b

   1. Initial program 74.0%

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

      1. inv-pow 74.0%

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

      2. difference-of-squares 87.8%

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

      3. unpow-prod-down 87.6%

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

      4. inv-pow 87.6%

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

      5. inv-pow 87.6%

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

   3. Applied egg-rr 87.6%

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

      1. associate-*r/ 87.7%

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

      2. *-rgt-identity 87.7%

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

      3. +-commutative 87.7%

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

   5. Simplified 87.7%

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

      1. pow1 87.7%

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

      2. frac-times 87.9%

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

      3. +-commutative 87.9%

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

      4. div-inv 87.9%

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

      5. +-commutative 87.9%

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

      6. inv-pow 87.9%

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

      7. inv-pow 87.9%

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

   7. Applied egg-rr 87.9%

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

      1. unpow1 87.9%

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

      2. associate-*l/ 99.5%

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

      3. unpow-1 99.5%

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

      4. unpow-1 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in a around 0 80.8%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 80.8%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

       2. *-commutative 80.8%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

       3. unpow2 80.8%

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

       4. associate-*r* 92.0%

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

       5. times-frac 92.6%

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

   12. Simplified 92.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \]

Alternative 5: 56.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. inv-pow 78.4%

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

   2. difference-of-squares 87.4%

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

   3. unpow-prod-down 88.1%

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

   4. inv-pow 88.1%

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

   5. inv-pow 88.1%

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

3. Applied egg-rr 88.1%

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

   1. associate-*r/ 88.2%

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

   2. *-rgt-identity 88.2%

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

   3. +-commutative 88.2%

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

5. Simplified 88.2%

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

6. Taylor expanded in a around 0 59.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 59.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

   2. times-frac 59.3%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{\pi}{{b}^{2}}} \]

   3. unpow2 59.3%

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

8. Simplified 59.3%

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

9. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2023279 
(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))))