NMSE Section 6.1 mentioned, B

Average Accuracy: 77.4% → 99.4%

Time: 17.6s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b + a}{\frac{\pi}{a}}}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{-\pi}{b}}{2}}{b + a}}{b - a}\\ \end{array} \]

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
 (if (<= a -2.3e+86)
   (* 0.5 (/ (/ PI (* b a)) a))
   (if (<= a 3e+106)
     (/ (/ 0.5 (/ (+ b a) (/ PI a))) b)
     (/ (/ (/ (/ (- PI) b) 2.0) (+ 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) {
	double tmp;
	if (a <= -2.3e+86) {
		tmp = 0.5 * ((((double) M_PI) / (b * a)) / a);
	} else if (a <= 3e+106) {
		tmp = (0.5 / ((b + a) / (((double) M_PI) / a))) / b;
	} else {
		tmp = (((-((double) M_PI) / b) / 2.0) / (b + a)) / (b - a);
	}
	return tmp;
}

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) {
	double tmp;
	if (a <= -2.3e+86) {
		tmp = 0.5 * ((Math.PI / (b * a)) / a);
	} else if (a <= 3e+106) {
		tmp = (0.5 / ((b + a) / (Math.PI / a))) / b;
	} else {
		tmp = (((-Math.PI / b) / 2.0) / (b + a)) / (b - a);
	}
	return tmp;
}

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):
	tmp = 0
	if a <= -2.3e+86:
		tmp = 0.5 * ((math.pi / (b * a)) / a)
	elif a <= 3e+106:
		tmp = (0.5 / ((b + a) / (math.pi / a))) / b
	else:
		tmp = (((-math.pi / b) / 2.0) / (b + a)) / (b - a)
	return tmp

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)
	tmp = 0.0
	if (a <= -2.3e+86)
		tmp = Float64(0.5 * Float64(Float64(pi / Float64(b * a)) / a));
	elseif (a <= 3e+106)
		tmp = Float64(Float64(0.5 / Float64(Float64(b + a) / Float64(pi / a))) / b);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-pi) / b) / 2.0) / Float64(b + a)) / Float64(b - a));
	end
	return tmp
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_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e+86)
		tmp = 0.5 * ((pi / (b * a)) / a);
	elseif (a <= 3e+106)
		tmp = (0.5 / ((b + a) / (pi / a))) / b;
	else
		tmp = (((-pi / b) / 2.0) / (b + a)) / (b - a);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[a, -2.3e+86], N[(0.5 * N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3e+106], N[(N[(0.5 / N[(N[(b + a), $MachinePrecision] / N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(N[((-Pi) / b), $MachinePrecision] / 2.0), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.2999999999999999e86

   1. Initial program 66.3%

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

   2. Simplified 66.4%

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

      [Start] 66.3	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*r/ [=>] 66.4	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-rgt-identity [=>] 66.4	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      sub-neg [=>] 66.4	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
      distribute-neg-frac [=>] 66.4	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
      metadata-eval [=>] 66.4	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Taylor expanded in b around 0 82.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]

   4. Simplified 98.4%

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

      [Start] 82.6	\[ 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]
      associate-*r/ [=>] 82.6	\[ \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]
      associate-/l* [=>] 82.6	\[ \color{blue}{\frac{0.5}{\frac{{a}^{2} \cdot b}{\pi}}} \]
      unpow2 [=>] 82.6	\[ \frac{0.5}{\frac{\color{blue}{\left(a \cdot a\right)} \cdot b}{\pi}} \]
      associate-*l* [=>] 98.4	\[ \frac{0.5}{\frac{\color{blue}{a \cdot \left(a \cdot b\right)}}{\pi}} \]

   5. Taylor expanded in a around 0 82.6%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{{a}^{2} \cdot b}{\pi}}} \]

   6. Simplified 98.5%

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

      [Start] 82.6	\[ \frac{0.5}{\frac{{a}^{2} \cdot b}{\pi}} \]
      unpow2 [=>] 82.6	\[ \frac{0.5}{\frac{\color{blue}{\left(a \cdot a\right)} \cdot b}{\pi}} \]
      associate-/l* [=>] 82.6	\[ \frac{0.5}{\color{blue}{\frac{a \cdot a}{\frac{\pi}{b}}}} \]
      associate-*r/ [<=] 98.4	\[ \frac{0.5}{\color{blue}{a \cdot \frac{a}{\frac{\pi}{b}}}} \]
      associate-/r/ [=>] 98.5	\[ \frac{0.5}{a \cdot \color{blue}{\left(\frac{a}{\pi} \cdot b\right)}} \]

   7. Applied egg-rr 99.1%

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

      [Start] 98.5	\[ \frac{0.5}{a \cdot \left(\frac{a}{\pi} \cdot b\right)} \]
      div-inv [=>] 98.5	\[ \color{blue}{0.5 \cdot \frac{1}{a \cdot \left(\frac{a}{\pi} \cdot b\right)}} \]
      *-commutative [=>] 98.5	\[ \color{blue}{\frac{1}{a \cdot \left(\frac{a}{\pi} \cdot b\right)} \cdot 0.5} \]
      associate-/l/ [<=] 99.1	\[ \color{blue}{\frac{\frac{1}{\frac{a}{\pi} \cdot b}}{a}} \cdot 0.5 \]
      associate-*l/ [=>] 99.1	\[ \frac{\frac{1}{\color{blue}{\frac{a \cdot b}{\pi}}}}{a} \cdot 0.5 \]
      associate-/l* [<=] 99.1	\[ \frac{\color{blue}{\frac{1 \cdot \pi}{a \cdot b}}}{a} \cdot 0.5 \]
      *-un-lft-identity [<=] 99.1	\[ \frac{\frac{\color{blue}{\pi}}{a \cdot b}}{a} \cdot 0.5 \]

   if -2.2999999999999999e86 < a < 3.0000000000000001e106

   1. Initial program 87.3%

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

   2. Simplified 87.4%

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

      [Start] 87.3	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l* [=>] 87.3	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]
      associate-*l/ [=>] 87.4	\[ \frac{\pi}{2} \cdot \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b \cdot b - a \cdot a}} \]
      *-lft-identity [=>] 87.4	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} - \frac{1}{b}}}{b \cdot b - a \cdot a} \]
      sub-neg [=>] 87.4	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} + \left(-\frac{1}{b}\right)}}{b \cdot b - a \cdot a} \]
      distribute-neg-frac [=>] 87.4	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b \cdot b - a \cdot a} \]
      metadata-eval [=>] 87.4	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b \cdot b - a \cdot a} \]

   3. Applied egg-rr 43.9%

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

      [Start] 87.4	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a} \]
      expm1-log1p-u [=>] 65.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\right)\right)} \]
      expm1-udef [=>] 43.9	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\right)} - 1} \]

   4. Simplified 98.8%

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

      [Start] 43.9	\[ e^{\mathsf{log1p}\left(\frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot 0.5\right)} - 1 \]
      expm1-def [=>] 77.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot 0.5\right)\right)} \]
      expm1-log1p [=>] 98.8	\[ \color{blue}{\frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot 0.5} \]
      *-commutative [=>] 98.8	\[ \color{blue}{0.5 \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]
      associate-*r* [=>] 98.8	\[ 0.5 \cdot \frac{\pi}{\color{blue}{\left(\left(a + b\right) \cdot a\right) \cdot b}} \]
      *-commutative [=>] 98.8	\[ 0.5 \cdot \frac{\pi}{\color{blue}{b \cdot \left(\left(a + b\right) \cdot a\right)}} \]
      *-commutative [=>] 98.8	\[ 0.5 \cdot \frac{\pi}{b \cdot \color{blue}{\left(a \cdot \left(a + b\right)\right)}} \]

   5. Applied egg-rr 99.4%

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

      [Start] 98.8	\[ 0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(a + b\right)\right)} \]
      associate-*r/ [=>] 98.8	\[ \color{blue}{\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot \left(a + b\right)\right)}} \]
      associate-/l/ [<=] 99.5	\[ \color{blue}{\frac{\frac{0.5 \cdot \pi}{a \cdot \left(a + b\right)}}{b}} \]
      associate-/l* [=>] 99.4	\[ \frac{\color{blue}{\frac{0.5}{\frac{a \cdot \left(a + b\right)}{\pi}}}}{b} \]
      *-commutative [=>] 99.4	\[ \frac{\frac{0.5}{\frac{\color{blue}{\left(a + b\right) \cdot a}}{\pi}}}{b} \]
      associate-/l* [=>] 99.4	\[ \frac{\frac{0.5}{\color{blue}{\frac{a + b}{\frac{\pi}{a}}}}}{b} \]
      +-commutative [=>] 99.4	\[ \frac{\frac{0.5}{\frac{\color{blue}{b + a}}{\frac{\pi}{a}}}}{b} \]

   if 3.0000000000000001e106 < a

   1. Initial program 59.6%

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

   2. Simplified 99.8%

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

      [Start] 59.6	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*r/ [=>] 59.6	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      difference-of-squares [=>] 80.0	\[ \frac{\frac{\pi}{2} \cdot 1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-/r* [=>] 80.8	\[ \color{blue}{\frac{\frac{\frac{\pi}{2} \cdot 1}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-rgt-identity [=>] 80.8	\[ \frac{\frac{\color{blue}{\frac{\pi}{2}}}{b + a}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l/ [=>] 99.8	\[ \color{blue}{\frac{\frac{\frac{\pi}{2}}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]

   3. Taylor expanded in b around 0 99.8%

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

   4. Simplified 99.8%

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

      [Start] 99.8	\[ \frac{\frac{\frac{-1 \cdot \frac{\pi}{b}}{2}}{b + a}}{b - a} \]
      associate-*r/ [=>] 99.8	\[ \frac{\frac{\frac{\color{blue}{\frac{-1 \cdot \pi}{b}}}{2}}{b + a}}{b - a} \]
      mul-1-neg [=>] 99.8	\[ \frac{\frac{\frac{\frac{\color{blue}{-\pi}}{b}}{2}}{b + a}}{b - a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b + a}{\frac{\pi}{a}}}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{-\pi}{b}}{2}}{b + a}}{b - a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	20160

\[\frac{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b + a}}{b - a} \]

Alternative 2
Accuracy	99.6%
Cost	7552

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

Alternative 3
Accuracy	73.5%
Cost	7442

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-12} \lor \neg \left(b \leq -1.26 \cdot 10^{-25}\right) \land \left(b \leq -1.22 \cdot 10^{-57} \lor \neg \left(b \leq 9.5 \cdot 10^{+31}\right)\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 4
Accuracy	80.6%
Cost	7442

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-11} \lor \neg \left(b \leq -1.26 \cdot 10^{-25}\right) \land \left(b \leq -1.22 \cdot 10^{-57} \lor \neg \left(b \leq 1.4 \cdot 10^{+32}\right)\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 5
Accuracy	80.6%
Cost	7441

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-25} \lor \neg \left(b \leq -1.22 \cdot 10^{-57}\right) \land b \leq 9.5 \cdot 10^{+31}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{b}\\ \end{array} \]

Alternative 6
Accuracy	80.4%
Cost	7440

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{if}\;b \leq -1 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{b}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 7
Accuracy	87.7%
Cost	7440

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ t_1 := t_0 \cdot \frac{0.5}{a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 8
Accuracy	87.7%
Cost	7440

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;b \leq -9.5 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+33}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 9
Accuracy	87.7%
Cost	7440

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \mathbf{elif}\;b \leq 10^{+33}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 10
Accuracy	87.7%
Cost	7440

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{b \cdot \left(b \cdot a\right)}{\pi}}\\ \end{array} \]

Alternative 11
Accuracy	87.7%
Cost	7440

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b \cdot a}{\pi}}}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{b \cdot \left(b \cdot a\right)}{\pi}}\\ \end{array} \]

Alternative 12
Accuracy	98.7%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -1.16 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{b \cdot a}\\ \end{array} \]

Alternative 13
Accuracy	99.0%
Cost	7304

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot -0.5}{b - a}\\ \end{array} \]

Alternative 14
Accuracy	99.4%
Cost	7304

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a}\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b + a}{\frac{\pi}{a}}}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot -0.5}{b - a}\\ \end{array} \]

Alternative 15
Accuracy	53.8%
Cost	6912

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

Reproduce

herbie shell --seed 2023138 
(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))))