NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.9% → 99.7%

Time: 12.0s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.5%

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

   1. *-commutative 78.5%

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

   2. associate-*r* 78.6%

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

   3. associate-*r/ 78.6%

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

   4. associate-*r* 78.6%

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

   5. *-rgt-identity 78.6%

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

   6. sub-neg 78.6%

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

   7. distribute-neg-frac 78.6%

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

   8. metadata-eval 78.6%

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

3. Simplified 78.6%

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

   1. *-un-lft-identity 78.6%

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

   2. difference-of-squares 88.4%

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

   3. times-frac 99.6%

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

   4. add-sqr-sqrt 49.8%

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

   5. sqrt-unprod 74.4%

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

   6. frac-times 74.4%

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

   7. metadata-eval 74.4%

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

   8. metadata-eval 74.4%

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

   9. frac-times 74.4%

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

   10. sqrt-unprod 32.2%

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

   11. add-sqr-sqrt 65.5%

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

   12. div-inv 65.5%

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

   13. metadata-eval 65.5%

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

6. Applied egg-rr 65.5%

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

   1. associate-*l/ 65.5%

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

   2. associate-/l* 65.5%

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

   3. +-commutative 65.5%

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

   4. *-commutative 65.5%

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

   5. +-commutative 65.5%

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

8. Simplified 65.5%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/r* 99.7%

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

    2. clear-num 99.6%

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

    3. inv-pow 99.6%

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

11. Applied egg-rr 99.6%

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

    1. unpow-1 99.6%

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

    2. associate-/r/ 99.7%

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

13. Simplified 99.7%

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

    1. associate-/r* 99.7%

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

    2. div-inv 99.6%

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

    3. clear-num 99.7%

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

15. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.7%

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

    2. *-rgt-identity 99.7%

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

17. Simplified 99.7%

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

18. Final simplification 99.7%

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

Alternative 2: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{-26} \lor \neg \left(a \leq -2.35 \cdot 10^{-61}\right) \land a \leq -7 \cdot 10^{-117}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

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 (or (<= a -1.1e-26) (and (not (<= a -2.35e-61)) (<= a -7e-117)))
     (* 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 <= -1.1e-26) || (!(a <= -2.35e-61) && (a <= -7e-117))) {
		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 <= -1.1e-26) || (!(a <= -2.35e-61) && (a <= -7e-117))) {
		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 <= -1.1e-26) or (not (a <= -2.35e-61) and (a <= -7e-117)):
		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(pi / Float64(b * a))
	tmp = 0.0
	if ((a <= -1.1e-26) || (!(a <= -2.35e-61) && (a <= -7e-117)))
		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 <= -1.1e-26) || (~((a <= -2.35e-61)) && (a <= -7e-117)))
		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[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, -1.1e-26], And[N[Not[LessEqual[a, -2.35e-61]], $MachinePrecision], LessEqual[a, -7e-117]]], 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 < -1.1e-26 or -2.3499999999999998e-61 < a < -6.9999999999999997e-117

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*r* 79.7%

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

      3. associate-*r/ 79.7%

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

      4. associate-*r* 79.7%

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

      5. *-rgt-identity 79.7%

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

      6. sub-neg 79.7%

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

      7. distribute-neg-frac 79.7%

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

      8. metadata-eval 79.7%

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

   3. Simplified 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. difference-of-squares 89.1%

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

      3. times-frac 99.5%

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

      4. add-sqr-sqrt 42.0%

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

      5. sqrt-unprod 62.7%

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

      6. frac-times 62.7%

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

      7. metadata-eval 62.7%

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

      8. metadata-eval 62.7%

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

      9. frac-times 62.7%

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

      10. sqrt-unprod 29.7%

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

      11. add-sqr-sqrt 51.2%

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

      12. div-inv 51.2%

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

      13. metadata-eval 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. associate-*l/ 51.2%

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

      2. associate-/l* 51.2%

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

      3. +-commutative 51.2%

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

      4. *-commutative 51.2%

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

      5. +-commutative 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in b around inf 99.6%

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

       1. associate-/r* 99.6%

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

       2. *-un-lft-identity 99.6%

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

       3. *-un-lft-identity 99.6%

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

       4. associate-/r* 99.6%

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

       5. *-commutative 99.6%

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

       6. associate-/r* 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

       2. associate-/l* 99.5%

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

       3. associate-/r* 99.5%

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

   13. Simplified 99.5%

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

   14. Taylor expanded in a around inf 84.4%

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

   if -1.1e-26 < a < -2.3499999999999998e-61 or -6.9999999999999997e-117 < a

   1. Initial program 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*r* 78.0%

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

      3. associate-*r/ 78.0%

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

      4. associate-*r* 78.0%

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

      5. *-rgt-identity 78.0%

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

      6. sub-neg 78.0%

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

      7. distribute-neg-frac 78.0%

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

      8. metadata-eval 78.0%

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

   3. Simplified 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. difference-of-squares 88.0%

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

      3. times-frac 99.7%

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

      4. add-sqr-sqrt 53.6%

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

      5. sqrt-unprod 80.3%

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

      6. frac-times 80.3%

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

      7. metadata-eval 80.3%

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

      8. metadata-eval 80.3%

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

      9. frac-times 80.3%

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

      10. sqrt-unprod 33.5%

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

      11. add-sqr-sqrt 72.6%

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

      12. div-inv 72.6%

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

      13. metadata-eval 72.6%

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

   6. Applied egg-rr 72.6%

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

      1. associate-*l/ 72.7%

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

      2. associate-/l* 72.7%

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

      3. +-commutative 72.7%

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

      4. *-commutative 72.7%

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

      5. +-commutative 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in b around inf 99.8%

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

       1. associate-/r* 99.8%

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

       2. *-un-lft-identity 99.8%

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

       3. *-un-lft-identity 99.8%

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

       4. associate-/r* 99.8%

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

       5. *-commutative 99.8%

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

       6. associate-/r* 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. *-lft-identity 99.8%

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

       2. associate-/l* 99.7%

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

       3. associate-/r* 99.7%

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

   13. Simplified 99.7%

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

   14. Taylor expanded in a around 0 73.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-26} \lor \neg \left(a \leq -2.35 \cdot 10^{-61}\right) \land a \leq -7 \cdot 10^{-117}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{b + a}}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+153)
   (* (/ PI (* b a)) (/ 0.5 a))
   (* (/ PI b) (/ (/ 0.5 (+ b a)) a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+153) {
		tmp = (((double) M_PI) / (b * a)) * (0.5 / a);
	} else {
		tmp = (((double) M_PI) / b) * ((0.5 / (b + a)) / a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+153) {
		tmp = (Math.PI / (b * a)) * (0.5 / a);
	} else {
		tmp = (Math.PI / b) * ((0.5 / (b + a)) / a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.5e+153:
		tmp = (math.pi / (b * a)) * (0.5 / a)
	else:
		tmp = (math.pi / b) * ((0.5 / (b + a)) / a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+153)
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(pi / b) * Float64(Float64(0.5 / Float64(b + a)) / a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.5e+153)
		tmp = (pi / (b * a)) * (0.5 / a);
	else
		tmp = (pi / b) * ((0.5 / (b + a)) / 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, -1.5e+153], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.50000000000000009e153

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

      2. associate-*r* 46.6%

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

      3. associate-*r/ 46.6%

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

      4. associate-*r* 46.6%

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

      5. *-rgt-identity 46.6%

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

      6. sub-neg 46.6%

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

      7. distribute-neg-frac 46.6%

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

      8. metadata-eval 46.6%

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

   3. Simplified 46.6%

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

      1. *-un-lft-identity 46.6%

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

      2. difference-of-squares 73.2%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 43.2%

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

      5. sqrt-unprod 75.6%

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

      6. frac-times 75.6%

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

      7. metadata-eval 75.6%

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

      8. metadata-eval 75.6%

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

      9. frac-times 75.6%

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

      10. sqrt-unprod 45.5%

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

      11. add-sqr-sqrt 72.8%

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

      12. div-inv 72.8%

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

      13. metadata-eval 72.8%

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

   6. Applied egg-rr 72.8%

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

      1. associate-*l/ 72.8%

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

      2. associate-/l* 72.8%

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

      3. +-commutative 72.8%

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

      4. *-commutative 72.8%

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

      5. +-commutative 72.8%

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

   8. Simplified 72.8%

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

   9. Taylor expanded in b around inf 99.9%

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

       1. associate-/r* 99.8%

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

       2. *-un-lft-identity 99.8%

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

       3. *-un-lft-identity 99.8%

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

       4. associate-/r* 99.9%

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

       5. *-commutative 99.9%

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

       6. associate-/r* 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. *-lft-identity 99.8%

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

       2. associate-/l* 99.8%

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

       3. associate-/r* 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in a around inf 99.8%

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

   if -1.50000000000000009e153 < a

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*r* 82.8%

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

      3. associate-*r/ 82.8%

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

      4. associate-*r* 82.8%

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

      5. *-rgt-identity 82.8%

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

      6. sub-neg 82.8%

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

      7. distribute-neg-frac 82.8%

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

      8. metadata-eval 82.8%

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

   3. Simplified 82.8%

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

      1. *-un-lft-identity 82.8%

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

      2. difference-of-squares 90.4%

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

      3. times-frac 99.6%

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

      4. add-sqr-sqrt 50.6%

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

      5. sqrt-unprod 74.3%

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

      6. frac-times 74.3%

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

      7. metadata-eval 74.3%

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

      8. metadata-eval 74.3%

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

      9. frac-times 74.3%

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

      10. sqrt-unprod 30.4%

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

      11. add-sqr-sqrt 64.6%

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

      12. div-inv 64.6%

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

      13. metadata-eval 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. associate-*l/ 64.6%

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

      2. associate-/l* 64.6%

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

      3. +-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. +-commutative 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in b around inf 99.7%

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

       1. associate-/r* 99.7%

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

       2. *-un-lft-identity 99.7%

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

       3. *-un-lft-identity 99.7%

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

       4. associate-/r* 99.7%

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

       5. *-commutative 99.7%

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

       6. associate-/r* 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-/r* 99.7%

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

   13. Simplified 99.7%

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

       1. *-commutative 99.7%

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

       2. associate-*r/ 99.6%

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

       3. associate-*l/ 99.5%

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

       4. *-commutative 99.5%

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

       5. times-frac 96.3%

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

   15. Applied egg-rr 96.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{b + a}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

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(pi / Float64(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[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. *-commutative 78.5%

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

   2. associate-*r* 78.6%

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

   3. associate-*r/ 78.6%

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

   4. associate-*r* 78.6%

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

   5. *-rgt-identity 78.6%

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

   6. sub-neg 78.6%

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

   7. distribute-neg-frac 78.6%

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

   8. metadata-eval 78.6%

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

3. Simplified 78.6%

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

   1. *-un-lft-identity 78.6%

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

   2. difference-of-squares 88.4%

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

   3. times-frac 99.6%

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

   4. add-sqr-sqrt 49.8%

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

   5. sqrt-unprod 74.4%

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

   6. frac-times 74.4%

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

   7. metadata-eval 74.4%

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

   8. metadata-eval 74.4%

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

   9. frac-times 74.4%

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

   10. sqrt-unprod 32.2%

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

   11. add-sqr-sqrt 65.5%

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

   12. div-inv 65.5%

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

   13. metadata-eval 65.5%

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

6. Applied egg-rr 65.5%

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

   1. associate-*l/ 65.5%

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

   2. associate-/l* 65.5%

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

   3. +-commutative 65.5%

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

   4. *-commutative 65.5%

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

   5. +-commutative 65.5%

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

8. Simplified 65.5%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/r* 99.7%

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

    2. *-un-lft-identity 99.7%

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

    3. *-un-lft-identity 99.7%

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

    4. associate-/r* 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-/r* 99.7%

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

11. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

    2. associate-/l* 99.6%

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

    3. associate-/r* 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 5: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.5%

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

   1. *-commutative 78.5%

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

   2. associate-*r* 78.6%

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

   3. associate-*r/ 78.6%

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

   4. associate-*r* 78.6%

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

   5. *-rgt-identity 78.6%

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

   6. sub-neg 78.6%

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

   7. distribute-neg-frac 78.6%

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

   8. metadata-eval 78.6%

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

3. Simplified 78.6%

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

   1. *-un-lft-identity 78.6%

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

   2. difference-of-squares 88.4%

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

   3. times-frac 99.6%

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

   4. add-sqr-sqrt 49.8%

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

   5. sqrt-unprod 74.4%

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

   6. frac-times 74.4%

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

   7. metadata-eval 74.4%

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

   8. metadata-eval 74.4%

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

   9. frac-times 74.4%

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

   10. sqrt-unprod 32.2%

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

   11. add-sqr-sqrt 65.5%

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

   12. div-inv 65.5%

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

   13. metadata-eval 65.5%

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

6. Applied egg-rr 65.5%

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

   1. associate-*l/ 65.5%

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

   2. associate-/l* 65.5%

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

   3. +-commutative 65.5%

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

   4. *-commutative 65.5%

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

   5. +-commutative 65.5%

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

8. Simplified 65.5%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/r* 99.7%

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

    2. *-un-lft-identity 99.7%

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

    3. *-un-lft-identity 99.7%

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

    4. associate-/r* 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-/r* 99.7%

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

11. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

    2. *-commutative 99.7%

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

    3. associate-/r* 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 6: 62.8% accurate, 2.3× speedup

Math

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

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(pi / Float64(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[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. *-commutative 78.5%

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

   2. associate-*r* 78.6%

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

   3. associate-*r/ 78.6%

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

   4. associate-*r* 78.6%

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

   5. *-rgt-identity 78.6%

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

   6. sub-neg 78.6%

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

   7. distribute-neg-frac 78.6%

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

   8. metadata-eval 78.6%

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

3. Simplified 78.6%

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

   1. *-un-lft-identity 78.6%

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

   2. difference-of-squares 88.4%

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

   3. times-frac 99.6%

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

   4. add-sqr-sqrt 49.8%

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

   5. sqrt-unprod 74.4%

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

   6. frac-times 74.4%

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

   7. metadata-eval 74.4%

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

   8. metadata-eval 74.4%

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

   9. frac-times 74.4%

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

   10. sqrt-unprod 32.2%

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

   11. add-sqr-sqrt 65.5%

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

   12. div-inv 65.5%

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

   13. metadata-eval 65.5%

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

6. Applied egg-rr 65.5%

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

   1. associate-*l/ 65.5%

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

   2. associate-/l* 65.5%

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

   3. +-commutative 65.5%

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

   4. *-commutative 65.5%

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

   5. +-commutative 65.5%

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

8. Simplified 65.5%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/r* 99.7%

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

    2. *-un-lft-identity 99.7%

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

    3. *-un-lft-identity 99.7%

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

    4. associate-/r* 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-/r* 99.7%

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

11. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

    2. associate-/l* 99.6%

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

    3. associate-/r* 99.6%

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

13. Simplified 99.6%

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

14. Taylor expanded in a around inf 61.7%

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

15. Final simplification 61.7%

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

Reproduce

herbie shell --seed 2024072 
(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))))