NMSE Section 6.1 mentioned, B

Average Accuracy: 77.1% → 99.5%

Time: 14.4s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

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

↓

(FPCore (a b) :precision binary64 (* PI (/ (/ 0.5 (+ b a)) (* b a))))

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));
}

↓

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

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));
}

↓

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

Python

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

↓

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

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

↓

function code(a, b)
	return Float64(pi * Float64(Float64(0.5 / Float64(b + a)) / Float64(b * a)))
end

MATLAB

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

↓

function tmp = code(a, b)
	tmp = pi * ((0.5 / (b + a)) / (b * a));
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]

↓

code[a_, b_] := N[(Pi * N[(N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

2. Simplified 77.1%

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

   [Start] 77.1	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*r/ [=>] 77.1	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-rgt-identity [=>] 77.1	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 85.0%

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

   [Start] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   *-commutative [=>] 77.1	\[ \color{blue}{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \]
   frac-2neg [=>] 77.1	\[ \left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \color{blue}{\frac{-\frac{\pi}{2}}{-\left(b \cdot b - a \cdot a\right)}} \]
   associate-*r/ [=>] 77.1	\[ \color{blue}{\frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(-\frac{\pi}{2}\right)}{-\left(b \cdot b - a \cdot a\right)}} \]
   div-inv [=>] 77.1	\[ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(-\color{blue}{\pi \cdot \frac{1}{2}}\right)}{-\left(b \cdot b - a \cdot a\right)} \]
   distribute-rgt-neg-in [=>] 77.1	\[ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \color{blue}{\left(\pi \cdot \left(-\frac{1}{2}\right)\right)}}{-\left(b \cdot b - a \cdot a\right)} \]
   metadata-eval [=>] 77.1	\[ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot \left(-\color{blue}{0.5}\right)\right)}{-\left(b \cdot b - a \cdot a\right)} \]
   metadata-eval [=>] 77.1	\[ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot \color{blue}{-0.5}\right)}{-\left(b \cdot b - a \cdot a\right)} \]
   difference-of-squares [=>] 85.0	\[ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot -0.5\right)}{-\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \]
   distribute-lft-neg-in [=>] 85.0	\[ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot -0.5\right)}{\color{blue}{\left(-\left(b + a\right)\right) \cdot \left(b - a\right)}} \]

4. Simplified 99.6%

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

   [Start] 85.0	\[ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot -0.5\right)}{\left(-\left(b + a\right)\right) \cdot \left(b - a\right)} \]
   *-commutative [=>] 85.0	\[ \frac{\color{blue}{\left(\pi \cdot -0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}}{\left(-\left(b + a\right)\right) \cdot \left(b - a\right)} \]
   times-frac [=>] 99.6	\[ \color{blue}{\frac{\pi \cdot -0.5}{-\left(b + a\right)} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a}} \]
   +-commutative [=>] 99.6	\[ \frac{\pi \cdot -0.5}{-\color{blue}{\left(a + b\right)}} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \]

5. Applied egg-rr 98.8%

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

   [Start] 99.6	\[ \frac{\pi \cdot -0.5}{-\left(a + b\right)} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \]
   clear-num [=>] 99.5	\[ \color{blue}{\frac{1}{\frac{-\left(a + b\right)}{\pi \cdot -0.5}}} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \]
   clear-num [=>] 99.5	\[ \frac{1}{\frac{-\left(a + b\right)}{\pi \cdot -0.5}} \cdot \color{blue}{\frac{1}{\frac{b - a}{\frac{1}{a} + \frac{-1}{b}}}} \]
   frac-times [=>] 98.7	\[ \color{blue}{\frac{1 \cdot 1}{\frac{-\left(a + b\right)}{\pi \cdot -0.5} \cdot \frac{b - a}{\frac{1}{a} + \frac{-1}{b}}}} \]
   metadata-eval [=>] 98.7	\[ \frac{\color{blue}{1}}{\frac{-\left(a + b\right)}{\pi \cdot -0.5} \cdot \frac{b - a}{\frac{1}{a} + \frac{-1}{b}}} \]
   neg-mul-1 [=>] 98.7	\[ \frac{1}{\frac{\color{blue}{-1 \cdot \left(a + b\right)}}{\pi \cdot -0.5} \cdot \frac{b - a}{\frac{1}{a} + \frac{-1}{b}}} \]
   *-commutative [=>] 98.7	\[ \frac{1}{\frac{-1 \cdot \left(a + b\right)}{\color{blue}{-0.5 \cdot \pi}} \cdot \frac{b - a}{\frac{1}{a} + \frac{-1}{b}}} \]
   times-frac [=>] 98.7	\[ \frac{1}{\color{blue}{\left(\frac{-1}{-0.5} \cdot \frac{a + b}{\pi}\right)} \cdot \frac{b - a}{\frac{1}{a} + \frac{-1}{b}}} \]
   metadata-eval [=>] 98.7	\[ \frac{1}{\left(\color{blue}{2} \cdot \frac{a + b}{\pi}\right) \cdot \frac{b - a}{\frac{1}{a} + \frac{-1}{b}}} \]
   frac-add [=>] 98.8	\[ \frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \frac{b - a}{\color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}}}} \]
   associate-/r/ [=>] 98.8	\[ \frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \color{blue}{\left(\frac{b - a}{1 \cdot b + a \cdot -1} \cdot \left(a \cdot b\right)\right)}} \]
   *-commutative [=>] 98.8	\[ \frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \left(\frac{b - a}{1 \cdot b + \color{blue}{-1 \cdot a}} \cdot \left(a \cdot b\right)\right)} \]
   neg-mul-1 [<=] 98.8	\[ \frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \left(\frac{b - a}{1 \cdot b + \color{blue}{\left(-a\right)}} \cdot \left(a \cdot b\right)\right)} \]
   *-un-lft-identity [<=] 98.8	\[ \frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \left(\frac{b - a}{\color{blue}{b} + \left(-a\right)} \cdot \left(a \cdot b\right)\right)} \]
   sub-neg [<=] 98.8	\[ \frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \left(\frac{b - a}{\color{blue}{b - a}} \cdot \left(a \cdot b\right)\right)} \]

6. Simplified 99.5%

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

   [Start] 98.8	\[ \frac{1}{\left(2 \cdot \frac{a + b}{\pi}\right) \cdot \left(\frac{b - a}{b - a} \cdot \left(a \cdot b\right)\right)} \]
   associate-/r* [=>] 99.5	\[ \color{blue}{\frac{\frac{1}{2 \cdot \frac{a + b}{\pi}}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)}} \]
   associate-/r* [=>] 99.5	\[ \frac{\color{blue}{\frac{\frac{1}{2}}{\frac{a + b}{\pi}}}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)} \]
   metadata-eval [=>] 99.5	\[ \frac{\frac{\color{blue}{0.5}}{\frac{a + b}{\pi}}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)} \]
   associate-/l* [<=] 99.6	\[ \frac{\color{blue}{\frac{0.5 \cdot \pi}{a + b}}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)} \]
   *-commutative [<=] 99.6	\[ \frac{\frac{\color{blue}{\pi \cdot 0.5}}{a + b}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)} \]
   *-rgt-identity [<=] 99.6	\[ \frac{\frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot 1}}{a + b}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)} \]
   associate-*r/ [<=] 99.5	\[ \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot \frac{1}{a + b}}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)} \]
   associate-*l* [=>] 99.5	\[ \frac{\color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}}{\frac{b - a}{b - a} \cdot \left(a \cdot b\right)} \]
   *-inverses [=>] 99.5	\[ \frac{\pi \cdot \left(0.5 \cdot \frac{1}{a + b}\right)}{\color{blue}{1} \cdot \left(a \cdot b\right)} \]
   times-frac [=>] 99.5	\[ \color{blue}{\frac{\pi}{1} \cdot \frac{0.5 \cdot \frac{1}{a + b}}{a \cdot b}} \]
   /-rgt-identity [=>] 99.5	\[ \color{blue}{\pi} \cdot \frac{0.5 \cdot \frac{1}{a + b}}{a \cdot b} \]
   associate-*r/ [=>] 99.5	\[ \pi \cdot \frac{\color{blue}{\frac{0.5 \cdot 1}{a + b}}}{a \cdot b} \]
   metadata-eval [=>] 99.5	\[ \pi \cdot \frac{\frac{\color{blue}{0.5}}{a + b}}{a \cdot b} \]
   +-commutative [=>] 99.5	\[ \pi \cdot \frac{\frac{0.5}{\color{blue}{b + a}}}{a \cdot b} \]
   *-commutative [=>] 99.5	\[ \pi \cdot \frac{\frac{0.5}{b + a}}{\color{blue}{b \cdot a}} \]

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	81.2%
Cost	7442

\[\begin{array}{l} \mathbf{if}\;b \leq -0.49 \lor \neg \left(b \leq -6.6 \cdot 10^{-45} \lor \neg \left(b \leq -1.12 \cdot 10^{-55}\right) \land b \leq 1.6 \cdot 10^{-48}\right):\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \end{array} \]

Alternative 2
Accuracy	88.4%
Cost	7442

\[\begin{array}{l} \mathbf{if}\;b \leq -13600 \lor \neg \left(b \leq -2.3 \cdot 10^{-44} \lor \neg \left(b \leq -1.12 \cdot 10^{-55}\right) \land b \leq 1.6 \cdot 10^{-48}\right):\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \end{array} \]

Alternative 3
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := \pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ t_1 := \pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;b \leq -28.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{b \cdot a}\\ \end{array} \]

Alternative 4
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := \pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ t_1 := \pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;b \leq -960000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{b}\\ \end{array} \]

Alternative 5
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := \pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;b \leq -4100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{b}\\ \end{array} \]

Alternative 6
Accuracy	88.1%
Cost	7440

\[\begin{array}{l} t_0 := \pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;b \leq -96000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{b}\\ \end{array} \]

Alternative 7
Accuracy	88.1%
Cost	7440

\[\begin{array}{l} t_0 := b \cdot \left(b \cdot a\right)\\ \mathbf{if}\;b \leq -680000000:\\ \;\;\;\;0.5 \cdot \frac{\pi}{t_0}\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\pi \cdot \frac{0.5}{t_0}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{b}\\ \end{array} \]

Alternative 8
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := b \cdot \left(b \cdot a\right)\\ \mathbf{if}\;b \leq -245000000:\\ \;\;\;\;0.5 \cdot \frac{\pi}{t_0}\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\pi \cdot \frac{0.5}{t_0}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{b}\\ \end{array} \]

Alternative 9
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} \mathbf{if}\;b \leq -118:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\frac{\pi}{b \cdot b}}}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{b}\\ \end{array} \]

Alternative 10
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} \mathbf{if}\;b \leq -45000000:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{\frac{0.5}{b}}}}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{b}\\ \end{array} \]

Alternative 11
Accuracy	53.0%
Cost	6912

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

Reproduce

herbie shell --seed 2023136 
(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))))