VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.4% → 99.7%

Time: 26.3s

Precision: binary64

Cost: 33160

Math

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 155000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

↓

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+83)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 155000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

↓

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+83) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 155000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

↓

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+83)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 155000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+83], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 155000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000029e83

   1. Initial program 41.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Simplified 66.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
      Step-by-step derivation

      [Start] 41.8	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 41.8	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 41.8	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 66.6	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 66.5	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 66.5	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 99.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -5.00000000000000029e83 < F < 1.55e8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
      Step-by-step derivation

      [Start] 99.5	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 99.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 99.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 99.6	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 99.6	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   if 1.55e8 < F

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Simplified 78.2%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
      Step-by-step derivation

      [Start] 54.4	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 54.4	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 54.4	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 78.1	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 78.1	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 78.1	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around inf 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 155000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	33160

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 155000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	33156

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\frac{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}{\frac{1}{\sin B}}}\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	26696

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 135000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Accuracy	99.6%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 6
Accuracy	99.0%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Accuracy	91.9%
Cost	14216

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-144}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.066:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 8
Accuracy	68.3%
Cost	14044

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \cos B \cdot \frac{-x}{\sin B}\\ \mathbf{if}\;F \leq -6.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 0.0007:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{B \cdot F} - t_0\\ \end{array} \]

Alternative 9
Accuracy	84.9%
Cost	14040

\[\begin{array}{l} t_0 := \cos B \cdot \frac{-x}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0044:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 8.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 10
Accuracy	84.9%
Cost	14040

\[\begin{array}{l} t_0 := \cos B \cdot \frac{-x}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0046:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-95}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot F\right) \cdot 0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 11
Accuracy	87.6%
Cost	14040

\[\begin{array}{l} t_0 := F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ t_1 := \cos B \cdot \frac{-x}{\sin B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot F\right) \cdot 0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_2\\ \end{array} \]

Alternative 12
Accuracy	77.1%
Cost	13912

\[\begin{array}{l} t_0 := \cos B \cdot \frac{-x}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0125:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -2.42 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{B \cdot F} - t_1\\ \end{array} \]

Alternative 13
Accuracy	60.5%
Cost	13780

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t_0\\ \mathbf{if}\;F \leq -6 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -0.0042:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{B \cdot F} - t_0\\ \end{array} \]

Alternative 14
Accuracy	91.7%
Cost	13772

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0102:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-144}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.066:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 15
Accuracy	55.9%
Cost	7508

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	54.5%
Cost	7244

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -2.45 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 17
Accuracy	60.2%
Cost	7244

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -0.0042:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 18
Accuracy	44.3%
Cost	7120

\[\begin{array}{l} \mathbf{if}\;F \leq -4.7 \cdot 10^{+188}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.45 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 19
Accuracy	43.6%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+188}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.45 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.52 \cdot 10^{-92}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 20
Accuracy	43.5%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Accuracy	43.5%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;F \leq -2.55 \cdot 10^{-37}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 22
Accuracy	43.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.52 \cdot 10^{-92}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + x \cdot \left(B \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 23
Accuracy	43.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-31}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 24
Accuracy	36.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -2.95 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 25
Accuracy	29.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+289}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 26
Accuracy	10.3%
Cost	192

\[\frac{-1}{B} \]

Reproduce

herbie shell --seed 2023164 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))