NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.9% → 99.6%

Time: 10.7s

Alternatives: 4

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: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-*r* 74.6%

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

   3. associate-*r/ 74.6%

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

   4. associate-*r* 74.6%

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

   5. *-rgt-identity 74.6%

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

   6. sub-neg 74.6%

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

   7. distribute-neg-frac 74.6%

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

   8. metadata-eval 74.6%

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

3. Simplified 74.6%

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

5. Taylor expanded in a around 0 53.8%

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

   1. associate-*r/ 53.8%

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

   2. *-commutative 53.8%

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

   3. metadata-eval 53.8%

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

   4. div-inv 53.8%

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

   5. *-un-lft-identity 53.8%

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

   6. associate-*l/ 53.7%

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

   7. *-un-lft-identity 53.7%

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

   8. difference-of-squares 58.0%

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

   9. times-frac 65.5%

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

   10. associate-*l/ 65.5%

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

   11. *-un-lft-identity 65.5%

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

   12. div-inv 65.5%

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

   13. metadata-eval 65.5%

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

   14. *-commutative 65.5%

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

   15. associate-*r/ 65.5%

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

7. Applied egg-rr 65.5%

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

   1. associate-*l/ 65.5%

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

   2. *-lft-identity 65.5%

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

   3. associate-/l* 65.5%

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

   4. associate-*l/ 65.5%

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

   5. +-commutative 65.5%

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

   6. associate-/l/ 65.5%

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

9. Simplified 65.5%

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

10. Taylor expanded in b around inf 99.5%

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

    1. frac-2neg 99.5%

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

    2. +-commutative 99.5%

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

    3. clear-num 99.5%

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

    4. frac-times 98.9%

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

    5. metadata-eval 98.9%

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

    6. metadata-eval 98.9%

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

12. Applied egg-rr 98.9%

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

    1. associate-/r* 99.6%

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

    2. distribute-neg-in 99.6%

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

    3. mul-1-neg 99.6%

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

    4. unsub-neg 99.6%

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

    5. mul-1-neg 99.6%

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

    6. *-commutative 99.6%

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

    7. associate-/l* 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 89.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;b \leq 1.96 \cdot 10^{-44}:\\ \;\;\;\;\frac{0.5}{a} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

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 (<= b 1.96e-44) (* (/ 0.5 a) t_0) (* 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 (b <= 1.96e-44) {
		tmp = (0.5 / a) * t_0;
	} 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 (b <= 1.96e-44) {
		tmp = (0.5 / a) * t_0;
	} 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 b <= 1.96e-44:
		tmp = (0.5 / a) * t_0
	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 (b <= 1.96e-44)
		tmp = Float64(Float64(0.5 / a) * t_0);
	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 (b <= 1.96e-44)
		tmp = (0.5 / a) * t_0;
	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[LessEqual[b, 1.96e-44], N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9599999999999999e-44

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-*r* 73.8%

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

      3. associate-*r/ 73.8%

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

      4. associate-*r* 73.8%

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

      5. *-rgt-identity 73.8%

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

      6. sub-neg 73.8%

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

      7. distribute-neg-frac 73.8%

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

      8. metadata-eval 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in a around 0 47.4%

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

      1. associate-*r/ 47.4%

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

      2. *-commutative 47.4%

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

      3. metadata-eval 47.4%

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

      4. div-inv 47.4%

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

      5. *-un-lft-identity 47.4%

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

      6. associate-*l/ 47.4%

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

      7. *-un-lft-identity 47.4%

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

      8. difference-of-squares 49.0%

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

      9. times-frac 55.1%

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

      10. associate-*l/ 55.1%

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

      11. *-un-lft-identity 55.1%

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

      12. div-inv 55.1%

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

      13. metadata-eval 55.1%

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

      14. *-commutative 55.1%

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

      15. associate-*r/ 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. associate-*l/ 55.2%

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

      2. *-lft-identity 55.2%

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

      3. associate-/l* 55.2%

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

      4. associate-*l/ 55.1%

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

      5. +-commutative 55.1%

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

      6. associate-/l/ 55.1%

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

   9. Simplified 55.1%

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

   10. Taylor expanded in b around inf 99.5%

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

   11. Taylor expanded in a around inf 72.7%

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

   if 1.9599999999999999e-44 < b

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*r* 76.7%

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

      3. associate-*r/ 76.8%

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

      4. associate-*r* 76.8%

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

      5. *-rgt-identity 76.8%

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

      6. sub-neg 76.8%

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

      7. distribute-neg-frac 76.8%

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

      8. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in a around 0 69.5%

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

      1. associate-*r/ 69.5%

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

      2. *-commutative 69.5%

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

      3. metadata-eval 69.5%

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

      4. div-inv 69.5%

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

      5. *-un-lft-identity 69.5%

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

      6. associate-*l/ 69.4%

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

      7. *-un-lft-identity 69.4%

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

      8. difference-of-squares 80.2%

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

      9. times-frac 91.0%

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

      10. associate-*l/ 90.9%

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

      11. *-un-lft-identity 90.9%

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

      12. div-inv 90.9%

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

      13. metadata-eval 90.9%

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

      14. *-commutative 90.9%

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

      15. associate-*r/ 90.9%

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

   7. Applied egg-rr 90.9%

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

      1. associate-*l/ 91.1%

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

      2. *-lft-identity 91.1%

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

      3. associate-/l* 91.1%

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

      4. associate-*l/ 90.9%

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

      5. +-commutative 90.9%

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

      6. associate-/l/ 91.0%

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

   9. Simplified 91.0%

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

   10. Taylor expanded in b around inf 99.6%

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

   11. Taylor expanded in a around 0 81.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.96 \cdot 10^{-44}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-*r* 74.6%

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

   3. associate-*r/ 74.6%

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

   4. associate-*r* 74.6%

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

   5. *-rgt-identity 74.6%

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

   6. sub-neg 74.6%

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

   7. distribute-neg-frac 74.6%

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

   8. metadata-eval 74.6%

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

3. Simplified 74.6%

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

5. Taylor expanded in a around 0 53.8%

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

   1. associate-*r/ 53.8%

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

   2. *-commutative 53.8%

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

   3. metadata-eval 53.8%

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

   4. div-inv 53.8%

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

   5. *-un-lft-identity 53.8%

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

   6. associate-*l/ 53.7%

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

   7. *-un-lft-identity 53.7%

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

   8. difference-of-squares 58.0%

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

   9. times-frac 65.5%

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

   10. associate-*l/ 65.5%

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

   11. *-un-lft-identity 65.5%

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

   12. div-inv 65.5%

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

   13. metadata-eval 65.5%

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

   14. *-commutative 65.5%

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

   15. associate-*r/ 65.5%

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

7. Applied egg-rr 65.5%

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

   1. associate-*l/ 65.5%

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

   2. *-lft-identity 65.5%

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

   3. associate-/l* 65.5%

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

   4. associate-*l/ 65.5%

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

   5. +-commutative 65.5%

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

   6. associate-/l/ 65.5%

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

9. Simplified 65.5%

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

10. Taylor expanded in b around inf 99.5%

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

    1. frac-times 99.0%

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

    2. *-commutative 99.0%

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

    3. +-commutative 99.0%

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

12. Applied egg-rr 99.0%

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

    1. *-commutative 99.0%

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

    2. times-frac 99.5%

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

    3. *-commutative 99.5%

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

    4. associate-*l/ 99.6%

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

    5. associate-/l* 99.6%

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

    6. associate-/r* 99.3%

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

    7. +-commutative 99.3%

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

14. Simplified 99.3%

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

15. Final simplification 99.3%

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

Alternative 4: 62.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-*r* 74.6%

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

   3. associate-*r/ 74.6%

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

   4. associate-*r* 74.6%

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

   5. *-rgt-identity 74.6%

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

   6. sub-neg 74.6%

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

   7. distribute-neg-frac 74.6%

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

   8. metadata-eval 74.6%

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

3. Simplified 74.6%

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

5. Taylor expanded in a around 0 53.8%

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

   1. associate-*r/ 53.8%

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

   2. *-commutative 53.8%

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

   3. metadata-eval 53.8%

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

   4. div-inv 53.8%

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

   5. *-un-lft-identity 53.8%

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

   6. associate-*l/ 53.7%

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

   7. *-un-lft-identity 53.7%

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

   8. difference-of-squares 58.0%

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

   9. times-frac 65.5%

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

   10. associate-*l/ 65.5%

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

   11. *-un-lft-identity 65.5%

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

   12. div-inv 65.5%

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

   13. metadata-eval 65.5%

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

   14. *-commutative 65.5%

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

   15. associate-*r/ 65.5%

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

7. Applied egg-rr 65.5%

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

   1. associate-*l/ 65.5%

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

   2. *-lft-identity 65.5%

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

   3. associate-/l* 65.5%

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

   4. associate-*l/ 65.5%

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

   5. +-commutative 65.5%

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

   6. associate-/l/ 65.5%

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

9. Simplified 65.5%

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

10. Taylor expanded in b around inf 99.5%

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

11. Taylor expanded in a around inf 64.8%

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

Reproduce

herbie shell --seed 2024180 
(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))))