NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 11.7s

Alternatives: 10

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

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+89}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 5e+89)
   (* (/ 0.5 a) (/ PI (* b (+ a b))))
   (* (/ PI b) (/ 0.5 (* a (- b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 5e+89) {
		tmp = (0.5 / a) * (((double) M_PI) / (b * (a + b)));
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * (b - a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+89) {
		tmp = (0.5 / a) * (Math.PI / (b * (a + b)));
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * (b - a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 5e+89:
		tmp = (0.5 / a) * (math.pi / (b * (a + b)))
	else:
		tmp = (math.pi / b) * (0.5 / (a * (b - a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 5e+89)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * Float64(b - a))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+89)
		tmp = (0.5 / a) * (pi / (b * (a + b)));
	else
		tmp = (pi / b) * (0.5 / (a * (b - a)));
	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[b, 5e+89], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999983e89

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-*r* 81.8%

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

      3. associate-*r/ 81.8%

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

      4. associate-/l* 81.8%

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

      5. /-rgt-identity 81.8%

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

      6. associate-/l* 81.8%

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

      7. difference-of-squares 89.6%

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

      8. associate-/l* 89.6%

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

      9. associate-/l* 99.5%

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

      10. associate-*r/ 91.0%

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

      11. sub-neg 91.0%

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

      12. distribute-neg-frac 91.0%

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

      13. metadata-eval 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in a around 0 61.4%

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

   5. Taylor expanded in b around inf 96.5%

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

      1. expm1-log1p-u 72.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a} \cdot \frac{0.5 \cdot \frac{\pi}{b}}{b + a}\right)\right)} \]

      2. expm1-udef 51.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \frac{0.5 \cdot \frac{\pi}{b}}{b + a}\right)} - 1} \]

      3. frac-times 51.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(0.5 \cdot \frac{\pi}{b}\right)}{a \cdot \left(b + a\right)}}\right)} - 1 \]

      4. *-un-lft-identity 51.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5 \cdot \frac{\pi}{b}}}{a \cdot \left(b + a\right)}\right)} - 1 \]

      5. times-frac 51.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{b + a}}\right)} - 1 \]

      6. +-commutative 51.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{\color{blue}{a + b}}\right)} - 1 \]

   7. Applied egg-rr 51.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a + b}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 72.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a + b}\right)\right)} \]

      2. expm1-log1p 96.5%

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

      3. associate-/l/ 95.9%

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

   9. Simplified 95.9%

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

   if 4.99999999999999983e89 < b

   1. Initial program 51.6%

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

      1. associate-*r/ 51.6%

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

      2. *-rgt-identity 51.6%

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

      3. associate-*l/ 51.6%

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

      4. difference-of-squares 75.1%

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

      5. *-commutative 75.1%

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

      6. times-frac 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. div-inv 99.8%

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

      3. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in a around 0 99.8%

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

      1. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. sqrt-unprod 75.2%

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

      4. sqr-neg 75.2%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 63.4%

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

      7. associate-/r* 63.4%

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

      8. associate-*r/ 63.4%

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

      9. associate-*l/ 63.4%

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

      10. *-commutative 63.4%

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

      11. times-frac 63.4%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 75.2%

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

      14. sqr-neg 75.2%

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

      15. sqrt-unprod 99.7%

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

      16. add-sqr-sqrt 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-/l/ 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+89}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot \left(b - a\right)}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{a + b}}{b - a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (/ (* (* PI 0.5) (/ (+ (/ 1.0 a) (/ -1.0 b)) (+ a b))) (- b a)))

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((pi * 0.5) * (((1.0 / a) + (-1.0 / b)) / (a + b))) / (b - a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*r/ 75.8%

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

   2. *-rgt-identity 75.8%

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

   3. associate-*l/ 75.8%

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

   4. difference-of-squares 86.7%

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

   5. *-commutative 86.7%

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

   6. times-frac 99.5%

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

   7. sub-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. associate-*l/ 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{1}{a} + \frac{-1}{b}}{a + b} \cdot \frac{\frac{\pi}{2}}{b - a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (* (/ (+ (/ 1.0 a) (/ -1.0 b)) (+ a b)) (/ (/ PI 2.0) (- b a))))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (((1.0 / a) + (-1.0 / b)) / (a + b)) * ((math.pi / 2.0) / (b - a))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(a + b)) * Float64(Float64(pi / 2.0) / Float64(b - a)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (((1.0 / a) + (-1.0 / b)) / (a + b)) * ((pi / 2.0) / (b - a));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / 2.0), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. associate-*r/ 75.8%

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

   2. *-rgt-identity 75.8%

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

   3. associate-*l/ 75.8%

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

   4. difference-of-squares 86.7%

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

   5. *-commutative 86.7%

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

   6. times-frac 99.5%

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

   7. sub-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 71.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{a \cdot b}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(-b\right)} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 5e+102)
   (/ (/ 0.5 a) (/ (* a b) PI))
   (* (/ PI (* a (- b))) (/ 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 5e+102) {
		tmp = (0.5 / a) / ((a * b) / ((double) M_PI));
	} else {
		tmp = (((double) M_PI) / (a * -b)) * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+102) {
		tmp = (0.5 / a) / ((a * b) / Math.PI);
	} else {
		tmp = (Math.PI / (a * -b)) * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 5e+102:
		tmp = (0.5 / a) / ((a * b) / math.pi)
	else:
		tmp = (math.pi / (a * -b)) * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 5e+102)
		tmp = Float64(Float64(0.5 / a) / Float64(Float64(a * b) / pi));
	else
		tmp = Float64(Float64(pi / Float64(a * Float64(-b))) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+102)
		tmp = (0.5 / a) / ((a * b) / pi);
	else
		tmp = (pi / (a * -b)) * (0.5 / 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[b, 5e+102], N[(N[(0.5 / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * (-b)), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e102

   1. Initial program 81.8%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

      3. associate-*l/ 81.8%

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

      4. difference-of-squares 89.6%

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

      5. *-commutative 89.6%

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

      6. times-frac 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around inf 67.0%

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

   5. Taylor expanded in b around 0 65.1%

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

      1. associate-*r/ 65.1%

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

   7. Simplified 65.1%

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

      1. associate-/l* 65.1%

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

      2. associate-/r* 65.1%

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

      3. frac-times 65.1%

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

   9. Applied egg-rr 65.1%

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

       1. associate-*r/ 65.1%

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

       2. metadata-eval 65.1%

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

       3. associate-*l/ 65.1%

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

       4. associate-/l* 65.1%

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

   11. Simplified 65.1%

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

   12. Taylor expanded in a around 0 65.1%

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

   if 5e102 < b

   1. Initial program 51.6%

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

      1. associate-*r/ 51.6%

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

      2. *-rgt-identity 51.6%

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

      3. associate-*l/ 51.6%

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

      4. difference-of-squares 75.1%

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

      5. *-commutative 75.1%

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

      6. times-frac 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 63.4%

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

      1. *-commutative 63.4%

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

      2. frac-2neg 63.4%

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

      3. metadata-eval 63.4%

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

      4. frac-times 63.4%

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

      5. *-un-lft-identity 63.4%

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

      6. div-inv 63.4%

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

      7. metadata-eval 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. times-frac 63.4%

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

      2. distribute-rgt-neg-in 63.4%

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

   8. Simplified 63.4%

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

   9. Taylor expanded in b around inf 63.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{a \cdot b}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(-b\right)} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 5: 93.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 / a) * (pi / (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 / a), $MachinePrecision] * N[(Pi / N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. associate-*r* 75.8%

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

   3. associate-*r/ 75.8%

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

   4. associate-/l* 75.8%

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

   5. /-rgt-identity 75.8%

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

   6. associate-/l* 75.8%

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

   7. difference-of-squares 86.7%

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

   8. associate-/l* 86.8%

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

   9. associate-/l* 99.5%

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

   10. associate-*r/ 88.2%

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

   11. sub-neg 88.2%

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

   12. distribute-neg-frac 88.2%

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

   13. metadata-eval 88.2%

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

3. Simplified 88.2%

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

4. Taylor expanded in a around 0 64.5%

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

5. Taylor expanded in b around inf 92.6%

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

   1. expm1-log1p-u 72.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a} \cdot \frac{0.5 \cdot \frac{\pi}{b}}{b + a}\right)\right)} \]

   2. expm1-udef 53.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \frac{0.5 \cdot \frac{\pi}{b}}{b + a}\right)} - 1} \]

   3. frac-times 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(0.5 \cdot \frac{\pi}{b}\right)}{a \cdot \left(b + a\right)}}\right)} - 1 \]

   4. *-un-lft-identity 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5 \cdot \frac{\pi}{b}}}{a \cdot \left(b + a\right)}\right)} - 1 \]

   5. times-frac 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{b + a}}\right)} - 1 \]

   6. +-commutative 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{\color{blue}{a + b}}\right)} - 1 \]

7. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a + b}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 72.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a + b}\right)\right)} \]

   2. expm1-log1p 92.6%

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

   3. associate-/l/ 91.8%

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

9. Simplified 91.8%

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

10. Final simplification 91.8%

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

Alternative 6: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\pi}{\left(a + b\right) \cdot \left(2 \cdot \left(a \cdot b\right)\right)} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ PI (* (+ a b) (* 2.0 (* a b)))))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.pi / ((a + b) * (2.0 * (a * b)))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(pi / Float64(Float64(a + b) * Float64(2.0 * Float64(a * b))))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = pi / ((a + b) * (2.0 * (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[(a + b), $MachinePrecision] * N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. associate-*r* 75.8%

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

   3. associate-*r/ 75.8%

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

   4. associate-/l* 75.8%

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

   5. /-rgt-identity 75.8%

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

   6. associate-/l* 75.8%

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

   7. difference-of-squares 86.7%

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

   8. associate-/l* 86.8%

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

   9. associate-/l* 99.5%

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

   10. associate-*r/ 88.2%

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

   11. sub-neg 88.2%

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

   12. distribute-neg-frac 88.2%

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

   13. metadata-eval 88.2%

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

3. Simplified 88.2%

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

4. Taylor expanded in a around 0 64.5%

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

   1. associate-*r/ 71.4%

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

   2. associate-/l/ 71.4%

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

   3. frac-times 71.5%

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

   4. *-un-lft-identity 71.5%

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

   5. +-commutative 71.5%

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

6. Applied egg-rr 71.5%

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

   1. associate-/l/ 71.3%

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

   2. *-commutative 71.3%

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

8. Simplified 71.3%

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

9. Taylor expanded in a around 0 99.2%

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

10. Final simplification 99.2%

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

Alternative 7: 31.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. associate-*r/ 75.8%

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

   2. *-rgt-identity 75.8%

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

   3. associate-*l/ 75.8%

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

   4. difference-of-squares 86.7%

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

   5. *-commutative 86.7%

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

   6. times-frac 99.5%

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

   7. sub-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a around inf 66.3%

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

5. Taylor expanded in b around 0 61.5%

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

   1. associate-*r/ 61.5%

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

7. Simplified 61.5%

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

   1. associate-/l* 61.5%

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

   2. frac-2neg 61.5%

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

   3. metadata-eval 61.5%

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

   4. distribute-rgt-neg-out 61.5%

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

   5. frac-times 61.2%

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

   6. metadata-eval 61.2%

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

   7. add-sqr-sqrt 29.9%

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

   8. sqrt-unprod 49.3%

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

   9. sqr-neg 49.3%

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

   10. sqrt-unprod 18.9%

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

   11. add-sqr-sqrt 35.3%

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

9. Applied egg-rr 35.3%

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

    1. add-sqr-sqrt 32.0%

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

    2. sqrt-unprod 45.5%

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

    3. frac-times 45.5%

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

    4. metadata-eval 45.5%

       \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\left(\frac{a}{\pi} \cdot \left(a \cdot b\right)\right) \cdot \left(\frac{a}{\pi} \cdot \left(a \cdot b\right)\right)}} \]

    5. metadata-eval 45.5%

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

    6. *-commutative 45.5%

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

    7. associate-*r* 45.5%

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

    8. associate-/r/ 45.5%

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

    9. *-commutative 45.5%

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

    10. associate-*r* 45.5%

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

    11. associate-/r/ 45.5%

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

    12. frac-times 45.5%

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

    13. associate-/l/ 45.5%

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

    14. associate-/l/ 45.5%

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

11. Applied egg-rr 35.3%

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

12. Final simplification 35.3%

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

Alternative 8: 64.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 / a) * (pi / (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 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.8%

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

   1. *-commutative 75.8%

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

   2. associate-*r* 75.8%

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

   3. associate-*r/ 75.8%

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

   4. associate-/l* 75.8%

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

   5. /-rgt-identity 75.8%

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

   6. associate-/l* 75.8%

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

   7. difference-of-squares 86.7%

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

   8. associate-/l* 86.8%

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

   9. associate-/l* 99.5%

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

   10. associate-*r/ 88.2%

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

   11. sub-neg 88.2%

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

   12. distribute-neg-frac 88.2%

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

   13. metadata-eval 88.2%

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

3. Simplified 88.2%

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

4. Taylor expanded in a around 0 64.5%

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

5. Taylor expanded in b around inf 92.6%

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

   1. expm1-log1p-u 72.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a} \cdot \frac{0.5 \cdot \frac{\pi}{b}}{b + a}\right)\right)} \]

   2. expm1-udef 53.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{a} \cdot \frac{0.5 \cdot \frac{\pi}{b}}{b + a}\right)} - 1} \]

   3. frac-times 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(0.5 \cdot \frac{\pi}{b}\right)}{a \cdot \left(b + a\right)}}\right)} - 1 \]

   4. *-un-lft-identity 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5 \cdot \frac{\pi}{b}}}{a \cdot \left(b + a\right)}\right)} - 1 \]

   5. times-frac 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{b + a}}\right)} - 1 \]

   6. +-commutative 53.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{\color{blue}{a + b}}\right)} - 1 \]

7. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a + b}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 72.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a + b}\right)\right)} \]

   2. expm1-log1p 92.6%

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

   3. associate-/l/ 91.8%

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

9. Simplified 91.8%

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

10. Taylor expanded in a around inf 61.5%

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

11. Final simplification 61.5%

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

Alternative 9: 64.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. associate-*r/ 75.8%

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

   2. *-rgt-identity 75.8%

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

   3. associate-*l/ 75.8%

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

   4. difference-of-squares 86.7%

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

   5. *-commutative 86.7%

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

   6. times-frac 99.5%

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

   7. sub-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a around inf 66.3%

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

5. Taylor expanded in b around 0 61.5%

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

   1. associate-*r/ 61.5%

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

7. Simplified 61.5%

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

   1. associate-/l* 61.5%

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

   2. associate-/r* 61.5%

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

   3. frac-times 61.5%

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

9. Applied egg-rr 61.5%

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

    1. associate-*r/ 61.5%

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

    2. metadata-eval 61.5%

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

    3. associate-*l/ 61.5%

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

    4. associate-/l* 61.5%

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

11. Simplified 61.5%

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

12. Final simplification 61.5%

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

Alternative 10: 64.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. associate-*r/ 75.8%

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

   2. *-rgt-identity 75.8%

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

   3. associate-*l/ 75.8%

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

   4. difference-of-squares 86.7%

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

   5. *-commutative 86.7%

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

   6. times-frac 99.5%

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

   7. sub-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a around inf 66.3%

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

5. Taylor expanded in b around 0 61.5%

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

   1. associate-*r/ 61.5%

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

7. Simplified 61.5%

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

   1. associate-/l* 61.5%

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

   2. associate-/r* 61.5%

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

   3. frac-times 61.5%

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

9. Applied egg-rr 61.5%

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

    1. associate-*r/ 61.5%

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

    2. metadata-eval 61.5%

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

    3. associate-*l/ 61.5%

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

    4. associate-/l* 61.5%

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

11. Simplified 61.5%

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

12. Taylor expanded in a around 0 61.5%

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

13. Final simplification 61.5%

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

Reproduce

herbie shell --seed 2023333 
(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))))