NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.2% → 99.7%

Time: 12.3s

Alternatives: 5

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. un-div-inv 80.7%

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

   2. difference-of-squares 88.1%

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

   3. associate-/r* 88.8%

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

   4. div-inv 88.8%

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

   5. metadata-eval 88.8%

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

4. Applied egg-rr 88.8%

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

   1. associate-*l/ 99.3%

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

   2. associate-/l* 99.2%

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

6. Applied egg-rr 99.2%

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

   1. associate-/l* 99.3%

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

   2. +-commutative 99.3%

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

   3. sub-neg 99.3%

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

   4. distribute-neg-frac 99.3%

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

   5. metadata-eval 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in a around 0 99.6%

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

    1. un-div-inv 99.7%

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

    2. associate-*r/ 99.7%

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

    3. +-commutative 99.7%

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

    4. clear-num 99.1%

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

    5. +-commutative 99.1%

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

    6. associate-*r/ 99.1%

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

11. Applied egg-rr 99.1%

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

    1. associate-/r/ 99.6%

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

    2. associate-*l/ 99.7%

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

    3. metadata-eval 99.7%

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

    4. distribute-lft-neg-in 99.7%

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

    5. neg-mul-1 99.7%

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

    6. remove-double-neg 99.7%

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

    7. associate-*r/ 99.7%

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

    8. *-commutative 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{+78}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.5e+78)
   (* PI (/ 0.5 (* a (* b (+ a b)))))
   (* (* 0.5 (/ PI b)) (/ (/ 1.0 a) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.5e+78) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.5e+78) {
		tmp = Math.PI * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = (0.5 * (Math.PI / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.5e+78:
		tmp = math.pi * (0.5 / (a * (b * (a + b))))
	else:
		tmp = (0.5 * (math.pi / b)) * ((1.0 / a) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.5e+78)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * Float64(a + b)))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.5e+78)
		tmp = pi * (0.5 / (a * (b * (a + b))));
	else
		tmp = (0.5 * (pi / b)) * ((1.0 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.5e+78], N[(Pi * N[(0.5 / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4999999999999999e78

   1. Initial program 83.8%

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

      1. un-div-inv 83.9%

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

      2. difference-of-squares 89.7%

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

      3. associate-/r* 90.6%

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

      4. div-inv 90.6%

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

      5. metadata-eval 90.6%

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

   4. Applied egg-rr 90.6%

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

      1. associate-*l/ 99.7%

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

      2. associate-/l* 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/l* 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-neg-frac 99.6%

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

      5. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.5%

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

       1. un-div-inv 99.6%

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

       2. associate-*r/ 99.7%

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

       3. +-commutative 99.7%

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

       4. clear-num 98.9%

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

       5. +-commutative 98.9%

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

       6. associate-*r/ 98.9%

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

   11. Applied egg-rr 98.9%

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

       1. associate-/r/ 99.5%

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

       2. associate-*l/ 99.6%

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

       3. metadata-eval 99.6%

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

       4. distribute-lft-neg-in 99.6%

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

       5. neg-mul-1 99.6%

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

       6. remove-double-neg 99.6%

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

       7. associate-/l* 99.7%

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

       8. associate-/r* 99.0%

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

       9. *-commutative 99.0%

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

       10. associate-*l* 94.5%

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

   13. Simplified 94.5%

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

   if 4.4999999999999999e78 < b

   1. Initial program 67.8%

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

      1. un-div-inv 67.9%

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

      2. difference-of-squares 81.6%

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

      3. associate-/r* 81.6%

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

      4. div-inv 81.6%

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

      5. metadata-eval 81.6%

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

   4. Applied egg-rr 81.6%

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

      1. associate-*l/ 97.8%

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

      2. associate-/l* 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. associate-/l* 97.8%

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

      2. +-commutative 97.8%

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

      3. sub-neg 97.8%

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

      4. distribute-neg-frac 97.8%

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

      5. metadata-eval 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in a around 0 99.7%

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

       1. associate-/r* 97.8%

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

   11. Simplified 97.8%

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

   12. Taylor expanded in a around 0 97.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\pi}{b}\right)} \cdot \frac{\frac{1}{a}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{+78}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+77}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e+77)
   (* PI (/ 0.5 (* a (* b (+ a b)))))
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.9e+77) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.9e+77) {
		tmp = Math.PI * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.9e+77:
		tmp = math.pi * (0.5 / (a * (b * (a + b))))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.9e+77)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * Float64(a + b)))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.9e+77)
		tmp = pi * (0.5 / (a * (b * (a + b))));
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.9e+77], N[(Pi * N[(0.5 / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9000000000000002e77

   1. Initial program 83.8%

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

      1. un-div-inv 83.9%

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

      2. difference-of-squares 89.7%

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

      3. associate-/r* 90.6%

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

      4. div-inv 90.6%

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

      5. metadata-eval 90.6%

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

   4. Applied egg-rr 90.6%

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

      1. associate-*l/ 99.7%

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

      2. associate-/l* 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/l* 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-neg-frac 99.6%

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

      5. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.5%

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

       1. un-div-inv 99.6%

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

       2. associate-*r/ 99.7%

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

       3. +-commutative 99.7%

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

       4. clear-num 98.9%

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

       5. +-commutative 98.9%

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

       6. associate-*r/ 98.9%

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

   11. Applied egg-rr 98.9%

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

       1. associate-/r/ 99.5%

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

       2. associate-*l/ 99.6%

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

       3. metadata-eval 99.6%

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

       4. distribute-lft-neg-in 99.6%

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

       5. neg-mul-1 99.6%

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

       6. remove-double-neg 99.6%

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

       7. associate-/l* 99.7%

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

       8. associate-/r* 99.0%

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

       9. *-commutative 99.0%

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

       10. associate-*l* 94.5%

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

   13. Simplified 94.5%

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

   if 2.9000000000000002e77 < b

   1. Initial program 67.8%

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

      1. un-div-inv 67.9%

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

      2. difference-of-squares 81.6%

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

      3. associate-/r* 81.6%

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

      4. div-inv 81.6%

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

      5. metadata-eval 81.6%

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

   4. Applied egg-rr 81.6%

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

      1. associate-*l/ 97.8%

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

      2. associate-/l* 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in b around inf 99.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+77}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. un-div-inv 80.7%

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

   2. difference-of-squares 88.1%

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

   3. associate-/r* 88.8%

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

   4. div-inv 88.8%

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

   5. metadata-eval 88.8%

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

4. Applied egg-rr 88.8%

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

   1. associate-*l/ 99.3%

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

   2. associate-/l* 99.2%

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

6. Applied egg-rr 99.2%

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

   1. associate-/l* 99.3%

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

   2. +-commutative 99.3%

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

   3. sub-neg 99.3%

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

   4. distribute-neg-frac 99.3%

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

   5. metadata-eval 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in a around 0 99.6%

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

    1. un-div-inv 99.7%

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

    2. associate-*r/ 99.7%

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

    3. +-commutative 99.7%

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

    4. clear-num 99.1%

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

    5. +-commutative 99.1%

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

    6. associate-*r/ 99.1%

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

11. Applied egg-rr 99.1%

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

    1. associate-/r/ 99.6%

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

    2. associate-*l/ 99.7%

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

    3. metadata-eval 99.7%

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

    4. distribute-lft-neg-in 99.7%

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

    5. neg-mul-1 99.7%

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

    6. remove-double-neg 99.7%

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

    7. associate-/l* 99.6%

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

    8. associate-/r* 99.1%

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

    9. *-commutative 99.1%

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

    10. associate-*l* 91.9%

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

13. Simplified 91.9%

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

14. Final simplification 91.9%

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

Alternative 5: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. un-div-inv 80.7%

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

   2. difference-of-squares 88.1%

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

   3. associate-/r* 88.8%

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

   4. div-inv 88.8%

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

   5. metadata-eval 88.8%

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

4. Applied egg-rr 88.8%

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

   1. associate-*l/ 99.3%

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

   2. associate-/l* 99.2%

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

6. Applied egg-rr 99.2%

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

   1. associate-/l* 99.3%

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

   2. +-commutative 99.3%

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

   3. sub-neg 99.3%

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

   4. distribute-neg-frac 99.3%

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

   5. metadata-eval 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in a around 0 99.6%

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

    1. *-commutative 99.6%

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

    2. associate-*r/ 99.6%

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

    3. +-commutative 99.6%

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

    4. frac-times 99.2%

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

    5. *-un-lft-identity 99.2%

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

    6. +-commutative 99.2%

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

11. Applied egg-rr 99.2%

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

12. Final simplification 99.2%

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

Reproduce

herbie shell --seed 2024059 
(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))))