NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.7% → 99.7%

Time: 8.5s

Alternatives: 7

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: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.9%

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

   4. difference-of-squares 91.9%

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

   5. times-frac 99.6%

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

   6. associate-/l/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around 0 99.6%

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

   1. associate-*l/ 99.6%

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

   2. associate-/r* 99.6%

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

   3. +-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-/l/ 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 90.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{b}}{a}\\ \mathbf{if}\;a \leq -4.35 \cdot 10^{-69}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI b) a)))
   (if (<= a -4.35e-69) (* t_0 (/ 0.5 a)) (* t_0 (/ 0.5 b)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = (((double) M_PI) / b) / a;
	double tmp;
	if (a <= -4.35e-69) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = (Math.PI / b) / a;
	double tmp;
	if (a <= -4.35e-69) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = (math.pi / b) / a
	tmp = 0
	if a <= -4.35e-69:
		tmp = t_0 * (0.5 / a)
	else:
		tmp = t_0 * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(Float64(pi / b) / a)
	tmp = 0.0
	if (a <= -4.35e-69)
		tmp = Float64(t_0 * Float64(0.5 / a));
	else
		tmp = Float64(t_0 * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = (pi / b) / a;
	tmp = 0.0;
	if (a <= -4.35e-69)
		tmp = t_0 * (0.5 / a);
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -4.35e-69], N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.34999999999999976e-69

   1. Initial program 88.0%

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

      1. associate-*r/ 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. associate-*l/ 87.9%

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

      4. difference-of-squares 92.6%

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

      5. times-frac 99.5%

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

      6. associate-/l/ 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 59.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 59.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 59.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 84.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   10. Applied egg-rr 59.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 84.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)\right)} \]

       2. expm1-log1p 99.3%

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

       3. associate-/l/ 99.7%

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

       4. *-rgt-identity 99.7%

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

       5. associate-*r/ 99.6%

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

       6. *-commutative 99.6%

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

       7. associate-/r/ 99.2%

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

       8. *-commutative 99.2%

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

       9. associate-*r/ 99.2%

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

       10. associate-/r* 99.2%

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

       11. metadata-eval 99.2%

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

       12. associate-/r/ 99.6%

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

       13. +-commutative 99.6%

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

       14. associate-/r* 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in a around inf 77.2%

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

   if -4.34999999999999976e-69 < a

   1. Initial program 80.3%

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

      1. associate-*r/ 80.4%

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

      2. *-rgt-identity 80.4%

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

      3. associate-*l/ 80.4%

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

      4. difference-of-squares 91.5%

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

      5. times-frac 99.6%

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

      6. associate-/l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 71.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 47.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 47.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 47.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 71.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   10. Applied egg-rr 47.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 70.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)\right)} \]

       2. expm1-log1p 99.2%

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

       3. associate-/l/ 99.7%

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

       4. *-rgt-identity 99.7%

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

       5. associate-*r/ 99.6%

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

       6. *-commutative 99.6%

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

       7. associate-/r/ 99.2%

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

       8. *-commutative 99.2%

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

       9. associate-*r/ 99.2%

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

       10. associate-/r* 99.2%

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

       11. metadata-eval 99.2%

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

       12. associate-/r/ 99.6%

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

       13. +-commutative 99.6%

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

       14. associate-/r* 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in a around 0 70.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.35 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{2}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4e-69) (/ (/ (/ PI a) 2.0) (* b a)) (* (/ (/ PI b) a) (/ 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4e-69) {
		tmp = ((((double) M_PI) / a) / 2.0) / (b * a);
	} else {
		tmp = ((((double) M_PI) / b) / a) * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4e-69) {
		tmp = ((Math.PI / a) / 2.0) / (b * a);
	} else {
		tmp = ((Math.PI / b) / a) * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4e-69:
		tmp = ((math.pi / a) / 2.0) / (b * a)
	else:
		tmp = ((math.pi / b) / a) * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4e-69)
		tmp = Float64(Float64(Float64(pi / a) / 2.0) / Float64(b * a));
	else
		tmp = Float64(Float64(Float64(pi / b) / a) * 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 (a <= -4e-69)
		tmp = ((pi / a) / 2.0) / (b * a);
	else
		tmp = ((pi / b) / a) * (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[a, -4e-69], N[(N[(N[(Pi / a), $MachinePrecision] / 2.0), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9999999999999999e-69

   1. Initial program 88.0%

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

      1. associate-*r/ 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. associate-*l/ 87.9%

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

      4. difference-of-squares 92.6%

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

      5. times-frac 99.5%

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

      6. associate-/l/ 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 59.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 59.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 59.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 84.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in b around 0 77.3%

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

   if -3.9999999999999999e-69 < a

   1. Initial program 80.3%

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

      1. associate-*r/ 80.4%

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

      2. *-rgt-identity 80.4%

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

      3. associate-*l/ 80.4%

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

      4. difference-of-squares 91.5%

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

      5. times-frac 99.6%

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

      6. associate-/l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 71.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 47.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 47.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 47.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 71.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   10. Applied egg-rr 47.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 70.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)\right)} \]

       2. expm1-log1p 99.2%

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

       3. associate-/l/ 99.7%

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

       4. *-rgt-identity 99.7%

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

       5. associate-*r/ 99.6%

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

       6. *-commutative 99.6%

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

       7. associate-/r/ 99.2%

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

       8. *-commutative 99.2%

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

       9. associate-*r/ 99.2%

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

       10. associate-/r* 99.2%

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

       11. metadata-eval 99.2%

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

       12. associate-/r/ 99.6%

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

       13. +-commutative 99.6%

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

       14. associate-/r* 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in a around 0 70.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{2}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 4: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{2}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{2}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e-69)
   (/ (/ (/ PI a) 2.0) (* b a))
   (/ (/ (/ PI b) 2.0) (* b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.3e-69) {
		tmp = ((((double) M_PI) / a) / 2.0) / (b * a);
	} else {
		tmp = ((((double) M_PI) / b) / 2.0) / (b * a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.3e-69) {
		tmp = ((Math.PI / a) / 2.0) / (b * a);
	} else {
		tmp = ((Math.PI / b) / 2.0) / (b * a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.3e-69:
		tmp = ((math.pi / a) / 2.0) / (b * a)
	else:
		tmp = ((math.pi / b) / 2.0) / (b * a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.3e-69)
		tmp = Float64(Float64(Float64(pi / a) / 2.0) / Float64(b * a));
	else
		tmp = Float64(Float64(Float64(pi / b) / 2.0) / Float64(b * a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.3e-69)
		tmp = ((pi / a) / 2.0) / (b * a);
	else
		tmp = ((pi / b) / 2.0) / (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[a, -3.3e-69], N[(N[(N[(Pi / a), $MachinePrecision] / 2.0), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / 2.0), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3e-69

   1. Initial program 88.0%

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

      1. associate-*r/ 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. associate-*l/ 87.9%

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

      4. difference-of-squares 92.6%

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

      5. times-frac 99.5%

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

      6. associate-/l/ 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 59.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 59.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 59.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 84.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in b around 0 77.3%

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

   if -3.3e-69 < a

   1. Initial program 80.3%

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

      1. associate-*r/ 80.4%

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

      2. *-rgt-identity 80.4%

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

      3. associate-*l/ 80.4%

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

      4. difference-of-squares 91.5%

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

      5. times-frac 99.6%

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

      6. associate-/l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 71.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 47.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 47.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 47.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 71.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in b around inf 70.1%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{2}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{2}}{b \cdot a}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.9%

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

   4. difference-of-squares 91.9%

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

   5. times-frac 99.6%

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

   6. associate-/l/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around 0 99.6%

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

   1. expm1-log1p-u 75.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 51.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

   3. un-div-inv 51.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

   4. +-commutative 51.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 75.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

   2. expm1-log1p 99.7%

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

   3. associate-/r* 99.7%

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

   4. +-commutative 99.7%

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

   5. *-commutative 99.7%

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

8. Simplified 99.7%

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

10. Applied egg-rr 51.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 75.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)\right)} \]

    2. expm1-log1p 99.2%

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

    3. associate-/l/ 99.7%

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

    4. *-rgt-identity 99.7%

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

    5. associate-*r/ 99.6%

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

    6. *-commutative 99.6%

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

    7. associate-/r/ 99.2%

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

    8. *-commutative 99.2%

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

    9. associate-*r/ 99.2%

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

    10. associate-/r* 99.2%

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

    11. metadata-eval 99.2%

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

    12. associate-/r/ 99.6%

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

    13. +-commutative 99.6%

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

    14. associate-/r* 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.9%

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

   4. difference-of-squares 91.9%

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

   5. times-frac 99.6%

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

   6. associate-/l/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around 0 99.6%

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

   1. expm1-log1p-u 75.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 51.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

   3. un-div-inv 51.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

   4. +-commutative 51.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 75.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

   2. expm1-log1p 99.7%

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

   3. associate-/r* 99.7%

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

   4. +-commutative 99.7%

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

   5. *-commutative 99.7%

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

8. Simplified 99.7%

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

10. Applied egg-rr 51.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 75.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)\right)} \]

    2. expm1-log1p 99.2%

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

    3. associate-/l/ 99.7%

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

    4. *-rgt-identity 99.7%

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

    5. associate-*r/ 99.6%

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

    6. *-commutative 99.6%

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

    7. associate-/r/ 99.2%

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

    8. *-commutative 99.2%

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

    9. associate-*r/ 99.2%

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

    10. associate-/r* 99.2%

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

    11. metadata-eval 99.2%

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

    12. associate-/r/ 99.6%

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

    13. +-commutative 99.6%

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

    14. associate-/r* 99.6%

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

12. Simplified 99.6%

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

    1. associate-/r* 99.6%

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

    2. associate-*r/ 99.6%

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

    3. +-commutative 99.6%

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

14. Applied egg-rr 99.6%

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

15. Final simplification 99.6%

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

Alternative 7: 63.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.9%

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

   4. difference-of-squares 91.9%

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

   5. times-frac 99.6%

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

   6. associate-/l/ 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around 0 99.6%

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

   1. expm1-log1p-u 75.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 51.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

   3. un-div-inv 51.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

   4. +-commutative 51.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 75.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

   2. expm1-log1p 99.7%

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

   3. associate-/r* 99.7%

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

   4. +-commutative 99.7%

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

   5. *-commutative 99.7%

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

8. Simplified 99.7%

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

10. Applied egg-rr 51.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 75.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(b \cdot a\right)}\right)\right)} \]

    2. expm1-log1p 99.2%

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

    3. associate-/l/ 99.7%

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

    4. *-rgt-identity 99.7%

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

    5. associate-*r/ 99.6%

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

    6. *-commutative 99.6%

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

    7. associate-/r/ 99.2%

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

    8. *-commutative 99.2%

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

    9. associate-*r/ 99.2%

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

    10. associate-/r* 99.2%

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

    11. metadata-eval 99.2%

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

    12. associate-/r/ 99.6%

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

    13. +-commutative 99.6%

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

    14. associate-/r* 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in a around inf 62.3%

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

14. Final simplification 62.3%

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

Reproduce

herbie shell --seed 2023320 
(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))))