NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 96.0%

Time: 10.0s

Precision: binary64

Cost: 8080

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ t_1 := \frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 0.5 (/ (/ PI (+ a b)) (- b a))) (+ (/ 1.0 a) (/ -1.0 b))))
        (t_1 (* (/ (/ 0.5 b) a) (/ PI a))))
   (if (<= a -1.3e+155)
     t_1
     (if (<= a -1.6e-206)
       t_0
       (if (<= a 1.95e-144)
         (/ (/ (/ PI b) (/ a 0.5)) b)
         (if (<= a 2.35e+58) t_0 t_1))))))

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 t_0 = (0.5 * ((((double) M_PI) / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	double t_1 = ((0.5 / b) / a) * (((double) M_PI) / a);
	double tmp;
	if (a <= -1.3e+155) {
		tmp = t_1;
	} else if (a <= -1.6e-206) {
		tmp = t_0;
	} else if (a <= 1.95e-144) {
		tmp = ((((double) M_PI) / b) / (a / 0.5)) / b;
	} else if (a <= 2.35e+58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 t_0 = (0.5 * ((Math.PI / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	double t_1 = ((0.5 / b) / a) * (Math.PI / a);
	double tmp;
	if (a <= -1.3e+155) {
		tmp = t_1;
	} else if (a <= -1.6e-206) {
		tmp = t_0;
	} else if (a <= 1.95e-144) {
		tmp = ((Math.PI / b) / (a / 0.5)) / b;
	} else if (a <= 2.35e+58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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):
	t_0 = (0.5 * ((math.pi / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b))
	t_1 = ((0.5 / b) / a) * (math.pi / a)
	tmp = 0
	if a <= -1.3e+155:
		tmp = t_1
	elif a <= -1.6e-206:
		tmp = t_0
	elif a <= 1.95e-144:
		tmp = ((math.pi / b) / (a / 0.5)) / b
	elif a <= 2.35e+58:
		tmp = t_0
	else:
		tmp = t_1
	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)
	t_0 = Float64(Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))) * Float64(Float64(1.0 / a) + Float64(-1.0 / b)))
	t_1 = Float64(Float64(Float64(0.5 / b) / a) * Float64(pi / a))
	tmp = 0.0
	if (a <= -1.3e+155)
		tmp = t_1;
	elseif (a <= -1.6e-206)
		tmp = t_0;
	elseif (a <= 1.95e-144)
		tmp = Float64(Float64(Float64(pi / b) / Float64(a / 0.5)) / b);
	elseif (a <= 2.35e+58)
		tmp = t_0;
	else
		tmp = t_1;
	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)
	t_0 = (0.5 * ((pi / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	t_1 = ((0.5 / b) / a) * (pi / a);
	tmp = 0.0;
	if (a <= -1.3e+155)
		tmp = t_1;
	elseif (a <= -1.6e-206)
		tmp = t_0;
	elseif (a <= 1.95e-144)
		tmp = ((pi / b) / (a / 0.5)) / b;
	elseif (a <= 2.35e+58)
		tmp = t_0;
	else
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.3e+155], t$95$1, If[LessEqual[a, -1.6e-206], t$95$0, If[LessEqual[a, 1.95e-144], N[(N[(N[(Pi / b), $MachinePrecision] / N[(a / 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, 2.35e+58], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3000000000000001e155 or 2.34999999999999986e58 < a

   1. Initial program 60.2%

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

   2. Simplified 82.9%

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

      [Start] 60.2%	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      times-frac [<=] 60.1%	\[ \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-commutative [<=] 60.1%	\[ \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      times-frac [=>] 60.1%	\[ \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      difference-of-squares [=>] 80.6%	\[ \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-/r* [=>] 82.9%	\[ \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      metadata-eval [=>] 82.9%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      sub-neg [=>] 82.9%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
      distribute-neg-frac [=>] 82.9%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
      metadata-eval [=>] 82.9%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Applied egg-rr 80.6%

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

      [Start] 82.9%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
      distribute-lft-in [=>] 82.9%	\[ \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b}} \]
      associate-/l/ [=>] 82.9%	\[ \left(\color{blue}{\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)}} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{-1}{b} \]
      associate-/l/ [=>] 80.6%	\[ \left(\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\color{blue}{\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)}} \cdot 0.5\right) \cdot \frac{-1}{b} \]

   4. Simplified 60.1%

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

      [Start] 80.6%	\[ \left(\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)} \cdot 0.5\right) \cdot \frac{-1}{b} \]
      distribute-lft-out [=>] 80.6%	\[ \color{blue}{\left(\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
      associate-*r* [<=] 80.6%	\[ \color{blue}{\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)} \]
      associate-*l/ [=>] 80.7%	\[ \color{blue}{\frac{\pi \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)}{\left(b - a\right) \cdot \left(b + a\right)}} \]
      *-commutative [<=] 80.7%	\[ \frac{\pi \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \]
      difference-of-squares [<=] 60.2%	\[ \frac{\pi \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)}{\color{blue}{b \cdot b - a \cdot a}} \]
      associate-*l/ [<=] 60.1%	\[ \color{blue}{\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)} \]
      distribute-lft-in [=>] 60.1%	\[ \frac{\pi}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{a} + 0.5 \cdot \frac{-1}{b}\right)} \]
      associate-*r/ [=>] 60.1%	\[ \frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{a}} + 0.5 \cdot \frac{-1}{b}\right) \]
      metadata-eval [=>] 60.1%	\[ \frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{\color{blue}{0.5}}{a} + 0.5 \cdot \frac{-1}{b}\right) \]
      associate-*r/ [=>] 60.1%	\[ \frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \color{blue}{\frac{0.5 \cdot -1}{b}}\right) \]
      metadata-eval [=>] 60.1%	\[ \frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{\color{blue}{-0.5}}{b}\right) \]

   5. Taylor expanded in b around 0 80.8%

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

   6. Simplified 99.8%

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

      [Start] 80.8%	\[ 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]
      *-commutative [=>] 80.8%	\[ \color{blue}{\frac{\pi}{{a}^{2} \cdot b} \cdot 0.5} \]
      associate-/r/ [<=] 80.8%	\[ \color{blue}{\frac{\pi}{\frac{{a}^{2} \cdot b}{0.5}}} \]
      associate-/l* [=>] 80.7%	\[ \frac{\pi}{\color{blue}{\frac{{a}^{2}}{\frac{0.5}{b}}}} \]
      associate-/l* [<=] 80.7%	\[ \color{blue}{\frac{\pi \cdot \frac{0.5}{b}}{{a}^{2}}} \]
      *-commutative [<=] 80.7%	\[ \frac{\color{blue}{\frac{0.5}{b} \cdot \pi}}{{a}^{2}} \]
      unpow2 [=>] 80.7%	\[ \frac{\frac{0.5}{b} \cdot \pi}{\color{blue}{a \cdot a}} \]
      times-frac [=>] 99.8%	\[ \color{blue}{\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}} \]

   if -1.3000000000000001e155 < a < -1.59999999999999988e-206 or 1.95000000000000007e-144 < a < 2.34999999999999986e58

   1. Initial program 93.3%

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

   2. Simplified 97.4%

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

      [Start] 93.3%	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      times-frac [<=] 93.4%	\[ \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-commutative [<=] 93.4%	\[ \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      times-frac [=>] 93.4%	\[ \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      difference-of-squares [=>] 96.2%	\[ \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-/r* [=>] 97.4%	\[ \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      metadata-eval [=>] 97.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      sub-neg [=>] 97.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
      distribute-neg-frac [=>] 97.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
      metadata-eval [=>] 97.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   if -1.59999999999999988e-206 < a < 1.95000000000000007e-144

   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. Simplified 76.4%

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

      [Start] 73.8%	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      times-frac [<=] 73.8%	\[ \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-commutative [<=] 73.8%	\[ \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      times-frac [=>] 73.8%	\[ \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      difference-of-squares [=>] 76.5%	\[ \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-/r* [=>] 76.4%	\[ \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      metadata-eval [=>] 76.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      sub-neg [=>] 76.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
      distribute-neg-frac [=>] 76.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
      metadata-eval [=>] 76.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Applied egg-rr 76.5%

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

      [Start] 76.4%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
      frac-add [=>] 76.5%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]
      *-un-lft-identity [<=] 76.5%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

   4. Simplified 76.5%

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

      [Start] 76.5%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + a \cdot -1}{a \cdot b} \]
      *-commutative [=>] 76.5%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]
      neg-mul-1 [<=] 76.5%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]
      sub-neg [<=] 76.5%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

   5. Applied egg-rr 76.4%

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

      [Start] 76.5%	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b - a}{a \cdot b} \]
      associate-*r/ [=>] 76.4%	\[ \color{blue}{\frac{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(b - a\right)}{a \cdot b}} \]
      *-commutative [=>] 76.4%	\[ \frac{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(b - a\right)}{\color{blue}{b \cdot a}} \]

   6. Simplified 76.3%

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

      [Start] 76.4%	\[ \frac{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(b - a\right)}{b \cdot a} \]
      associate-*l* [=>] 76.4%	\[ \frac{\color{blue}{\frac{\frac{\pi}{b + a}}{b - a} \cdot \left(0.5 \cdot \left(b - a\right)\right)}}{b \cdot a} \]
      associate-/l/ [=>] 76.3%	\[ \frac{\color{blue}{\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)}} \cdot \left(0.5 \cdot \left(b - a\right)\right)}{b \cdot a} \]

   7. Taylor expanded in b around inf 98.1%

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

   8. Applied egg-rr 42.8%

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

      [Start] 98.1%	\[ \frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a} \]
      expm1-log1p-u [=>] 71.1%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\right)\right)} \]
      expm1-udef [=>] 42.8%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\right)} - 1} \]
      times-frac [=>] 42.8%	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}}\right)} - 1 \]

   9. Simplified 98.3%

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

      [Start] 42.8%	\[ e^{\mathsf{log1p}\left(\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\right)} - 1 \]
      expm1-def [=>] 71.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\right)\right)} \]
      expm1-log1p [=>] 98.3%	\[ \color{blue}{\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}} \]
      associate-*l/ [=>] 98.3%	\[ \color{blue}{\frac{0.5 \cdot \frac{\frac{\pi}{b}}{a}}{b}} \]
      associate-*r/ [=>] 98.3%	\[ \frac{\color{blue}{\frac{0.5 \cdot \frac{\pi}{b}}{a}}}{b} \]
      *-commutative [=>] 98.3%	\[ \frac{\frac{\color{blue}{\frac{\pi}{b} \cdot 0.5}}{a}}{b} \]
      associate-/l* [=>] 98.3%	\[ \frac{\color{blue}{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-206}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.0%
Cost	8080

\[\begin{array}{l} t_0 := \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ t_1 := \frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	95.6%
Cost	7952

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ t_1 := \frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{if}\;a \leq -5 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	88.1%
Cost	7560

\[\begin{array}{l} t_0 := \frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{-100}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{-1}{b}\\ \mathbf{elif}\;a \leq 3.65 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	87.9%
Cost	7432

\[\begin{array}{l} t_0 := \frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{if}\;a \leq -6.7 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-98}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{-0.5}{b \cdot b - a \cdot a}\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	88.1%
Cost	7432

\[\begin{array}{l} t_0 := \frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{if}\;a \leq -1 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot -0.5}{b \cdot b - a \cdot a}\\ \mathbf{elif}\;a \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	88.2%
Cost	7432

\[\begin{array}{l} t_0 := \frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\pi \cdot \frac{-0.5}{b}}{b \cdot b - a \cdot a}\\ \mathbf{elif}\;a \leq 0.0037:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	80.2%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-99} \lor \neg \left(a \leq 1.55 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\\ \end{array} \]

Alternative 8
Accuracy	85.6%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-98} \lor \neg \left(a \leq 1.35 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\\ \end{array} \]

Alternative 9
Accuracy	85.6%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-98} \lor \neg \left(a \leq 1.12 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{\frac{a}{0.5}}}{b}\\ \end{array} \]

Alternative 10
Accuracy	79.9%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-98}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot a}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}\\ \end{array} \]

Alternative 11
Accuracy	58.1%
Cost	6912

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

Reproduce

herbie shell --seed 2023167 
(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))))