NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 11.9s

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: 78.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{a + b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (* (/ PI a) (/ 0.5 b)) (+ a b)))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(pi / a) * Float64(0.5 / b)) / Float64(a + b))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((pi / a) * (0.5 / b)) / (a + b);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 74.0%

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

   2. associate-*r* 73.9%

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

   3. associate-*r/ 74.0%

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

   4. associate-*r* 74.0%

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

   5. *-rgt-identity 74.0%

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

   6. sub-neg 74.0%

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

   7. distribute-neg-frac 74.0%

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

   8. metadata-eval 74.0%

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

3. Simplified 74.0%

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

   1. *-commutative 74.0%

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

   2. difference-of-squares 84.1%

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

   3. times-frac 99.6%

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

   4. div-inv 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 44.7%

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

   7. sqrt-unprod 71.3%

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

   8. frac-times 71.3%

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

   9. metadata-eval 71.3%

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

   10. metadata-eval 71.3%

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

   11. frac-times 71.3%

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

   12. sqrt-unprod 33.9%

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

   13. add-sqr-sqrt 65.2%

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

6. Applied egg-rr 65.2%

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

   1. associate-*l/ 65.2%

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

   2. *-commutative 65.2%

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

   3. +-commutative 65.2%

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

   4. +-commutative 65.2%

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

8. Simplified 65.2%

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

9. Taylor expanded in b around inf 61.5%

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

    1. distribute-lft-out 61.5%

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

    2. neg-mul-1 61.5%

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

    3. sub-neg 61.5%

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

    4. remove-double-neg 61.5%

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

11. Simplified 61.5%

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

12. Taylor expanded in a around 0 99.7%

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

    1. *-commutative 99.7%

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

    2. associate-*l/ 99.7%

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

    3. times-frac 99.7%

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

14. Simplified 99.7%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.15e+154)
   (* 0.5 (/ PI (* (+ a b) (* a b))))
   (* 0.5 (/ (/ PI b) (* a (+ a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.15e+154) {
		tmp = 0.5 * (((double) M_PI) / ((a + b) * (a * b)));
	} else {
		tmp = 0.5 * ((((double) M_PI) / b) / (a * (a + b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.15e+154) {
		tmp = 0.5 * (Math.PI / ((a + b) * (a * b)));
	} else {
		tmp = 0.5 * ((Math.PI / b) / (a * (a + b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.15e+154:
		tmp = 0.5 * (math.pi / ((a + b) * (a * b)))
	else:
		tmp = 0.5 * ((math.pi / b) / (a * (a + b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.15e+154)
		tmp = Float64(0.5 * Float64(pi / Float64(Float64(a + b) * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / b) / Float64(a * Float64(a + b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.15e+154)
		tmp = 0.5 * (pi / ((a + b) * (a * b)));
	else
		tmp = 0.5 * ((pi / b) / (a * (a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.15e+154], N[(0.5 * N[(Pi / N[(N[(a + b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(a * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e154

   1. Initial program 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*r* 51.5%

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

      3. associate-*r/ 51.5%

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

      4. associate-*r* 51.5%

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

      5. *-rgt-identity 51.5%

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

      6. sub-neg 51.5%

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

      7. distribute-neg-frac 51.5%

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

      8. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

      1. *-commutative 51.5%

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

      2. difference-of-squares 67.8%

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

      3. times-frac 99.7%

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

      4. div-inv 99.7%

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

      5. metadata-eval 99.7%

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

      6. add-sqr-sqrt 44.1%

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

      7. sqrt-unprod 68.1%

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

      8. frac-times 68.1%

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

      9. metadata-eval 68.1%

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

      10. metadata-eval 68.1%

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

      11. frac-times 68.1%

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

      12. sqrt-unprod 35.9%

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

      13. add-sqr-sqrt 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. associate-*l/ 67.3%

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

      2. *-commutative 67.3%

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

      3. +-commutative 67.3%

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

      4. +-commutative 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in b around inf 99.9%

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

       1. associate-/l* 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. associate-/l/ 98.2%

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

   13. Simplified 98.2%

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

   if -1.15e154 < a

   1. Initial program 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-*r* 78.4%

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

      3. associate-*r/ 78.5%

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

      4. associate-*r* 78.5%

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

      5. *-rgt-identity 78.5%

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

      6. sub-neg 78.5%

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

      7. distribute-neg-frac 78.5%

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

      8. metadata-eval 78.5%

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

   3. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. difference-of-squares 87.4%

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

      3. times-frac 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. add-sqr-sqrt 44.9%

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

      7. sqrt-unprod 71.9%

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

      8. frac-times 71.9%

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

      9. metadata-eval 71.9%

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

      10. metadata-eval 71.9%

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

      11. frac-times 71.9%

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

      12. sqrt-unprod 33.5%

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

      13. add-sqr-sqrt 64.8%

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

   6. Applied egg-rr 64.8%

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

      1. associate-*l/ 64.8%

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

      2. *-commutative 64.8%

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

      3. +-commutative 64.8%

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

      4. +-commutative 64.8%

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

   8. Simplified 64.8%

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

   9. Taylor expanded in b around inf 99.7%

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

       1. associate-/l* 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. associate-/l/ 97.7%

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

   13. Simplified 97.7%

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

       1. *-un-lft-identity 97.7%

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

       2. associate-*r* 94.0%

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

       3. times-frac 95.5%

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

   15. Applied egg-rr 95.5%

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

       1. associate-*l/ 95.6%

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

       2. *-lft-identity 95.6%

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

       3. *-commutative 95.6%

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

   17. Simplified 95.6%

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

Alternative 3: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{0.5 \cdot \frac{\pi}{a \cdot b}}{a + b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (* 0.5 (/ PI (* a b))) (+ a b)))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(a + b))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 * (pi / (a * b))) / (a + b);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 74.0%

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

   2. associate-*r* 73.9%

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

   3. associate-*r/ 74.0%

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

   4. associate-*r* 74.0%

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

   5. *-rgt-identity 74.0%

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

   6. sub-neg 74.0%

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

   7. distribute-neg-frac 74.0%

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

   8. metadata-eval 74.0%

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

3. Simplified 74.0%

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

   1. *-commutative 74.0%

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

   2. difference-of-squares 84.1%

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

   3. times-frac 99.6%

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

   4. div-inv 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 44.7%

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

   7. sqrt-unprod 71.3%

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

   8. frac-times 71.3%

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

   9. metadata-eval 71.3%

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

   10. metadata-eval 71.3%

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

   11. frac-times 71.3%

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

   12. sqrt-unprod 33.9%

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

   13. add-sqr-sqrt 65.2%

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

6. Applied egg-rr 65.2%

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

   1. associate-*l/ 65.2%

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

   2. *-commutative 65.2%

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

   3. +-commutative 65.2%

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

   4. +-commutative 65.2%

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

8. Simplified 65.2%

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

9. Taylor expanded in b around inf 99.7%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \pi \cdot \frac{\frac{0.5}{a + b}}{a \cdot b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* PI (/ (/ 0.5 (+ a b)) (* a b))))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(pi * Float64(Float64(0.5 / Float64(a + b)) / Float64(a * b)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = pi * ((0.5 / (a + b)) / (a * b));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(Pi * N[(N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 74.0%

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

   2. associate-*r* 73.9%

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

   3. associate-*r/ 74.0%

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

   4. associate-*r* 74.0%

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

   5. *-rgt-identity 74.0%

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

   6. sub-neg 74.0%

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

   7. distribute-neg-frac 74.0%

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

   8. metadata-eval 74.0%

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

3. Simplified 74.0%

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

   1. *-commutative 74.0%

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

   2. difference-of-squares 84.1%

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

   3. times-frac 99.6%

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

   4. div-inv 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 44.7%

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

   7. sqrt-unprod 71.3%

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

   8. frac-times 71.3%

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

   9. metadata-eval 71.3%

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

   10. metadata-eval 71.3%

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

   11. frac-times 71.3%

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

   12. sqrt-unprod 33.9%

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

   13. add-sqr-sqrt 65.2%

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

6. Applied egg-rr 65.2%

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

   1. associate-*l/ 65.2%

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

   2. *-commutative 65.2%

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

   3. +-commutative 65.2%

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

   4. +-commutative 65.2%

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

8. Simplified 65.2%

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

9. Taylor expanded in b around inf 61.5%

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

    1. distribute-lft-out 61.5%

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

    2. neg-mul-1 61.5%

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

    3. sub-neg 61.5%

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

    4. remove-double-neg 61.5%

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

11. Simplified 61.5%

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

12. Taylor expanded in a around 0 99.7%

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

    1. *-commutative 99.7%

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

    2. associate-*l/ 99.7%

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

    3. times-frac 99.7%

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

14. Simplified 99.7%

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

    1. div-inv 99.6%

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

    2. frac-times 99.7%

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

    3. associate-*l/ 99.7%

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

    4. associate-*l* 99.7%

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

    5. div-inv 99.7%

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

    6. div-inv 99.6%

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

    7. associate-*l* 99.6%

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

16. Applied egg-rr 99.6%

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

    1. associate-*l/ 99.6%

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

    2. *-lft-identity 99.6%

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

18. Simplified 99.6%

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

Alternative 5: 98.9% accurate, 1.9× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* 0.5 (/ PI (* (+ a b) (* a b)))))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(0.5 * Float64(pi / Float64(Float64(a + b) * Float64(a * b))))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = 0.5 * (pi / ((a + b) * (a * b)));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(0.5 * N[(Pi / N[(N[(a + b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 74.0%

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

   2. associate-*r* 73.9%

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

   3. associate-*r/ 74.0%

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

   4. associate-*r* 74.0%

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

   5. *-rgt-identity 74.0%

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

   6. sub-neg 74.0%

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

   7. distribute-neg-frac 74.0%

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

   8. metadata-eval 74.0%

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

3. Simplified 74.0%

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

   1. *-commutative 74.0%

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

   2. difference-of-squares 84.1%

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

   3. times-frac 99.6%

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

   4. div-inv 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 44.7%

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

   7. sqrt-unprod 71.3%

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

   8. frac-times 71.3%

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

   9. metadata-eval 71.3%

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

   10. metadata-eval 71.3%

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

   11. frac-times 71.3%

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

   12. sqrt-unprod 33.9%

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

   13. add-sqr-sqrt 65.2%

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

6. Applied egg-rr 65.2%

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

   1. associate-*l/ 65.2%

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

   2. *-commutative 65.2%

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

   3. +-commutative 65.2%

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

   4. +-commutative 65.2%

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

8. Simplified 65.2%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/l* 99.7%

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

11. Applied egg-rr 99.7%

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

    1. associate-/l/ 97.8%

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

13. Simplified 97.8%

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

Reproduce

herbie shell --seed 2024084 
(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))))