VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.2% → 99.6%

Time: 18.3s

Precision: binary64

Cost: 26696

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

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 -3.5e+27)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (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 <= -3.5e+27) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / 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 <= -3.5e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / 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, -3.5e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $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 < -3.5000000000000002e27

   1. Initial program 54.6%

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

   2. Simplified 66.8%

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

      [Start] 54.6%	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 54.6%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 54.6%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 66.7%	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 66.7%	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 66.7%	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in x around 0 66.8%

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

   4. Simplified 66.8%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
      Step-by-step derivation

      [Start] 66.8%	\[ F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{{F}^{2} + 2}}\right) - \frac{x}{\tan B} \]
      associate-*l/ [=>] 66.8%	\[ F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{{F}^{2} + 2}}}{\sin B}} - \frac{x}{\tan B} \]
      *-lft-identity [=>] 66.8%	\[ F \cdot \frac{\color{blue}{\sqrt{\frac{1}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [=>] 66.8%	\[ F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]
      fma-udef [<=] 66.8%	\[ F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   5. Taylor expanded in F around -inf 99.8%

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

   if -3.5000000000000002e27 < F < 1e8

   1. Initial program 99.3%

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

   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] 99.3%	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 99.3%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 99.3%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 99.4%	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 99.4%	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 99.4%	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in x around 0 99.6%

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

   4. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
      Step-by-step derivation

      [Start] 99.6%	\[ F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{{F}^{2} + 2}}\right) - \frac{x}{\tan B} \]
      associate-*l/ [=>] 99.6%	\[ F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{{F}^{2} + 2}}}{\sin B}} - \frac{x}{\tan B} \]
      *-lft-identity [=>] 99.6%	\[ F \cdot \frac{\color{blue}{\sqrt{\frac{1}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [=>] 99.6%	\[ F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]
      fma-udef [<=] 99.6%	\[ F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   if 1e8 < F

   1. Initial program 59.1%

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

   2. Simplified 71.3%

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

      [Start] 59.1%	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 59.1%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 59.1%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 71.2%	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 71.1%	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 71.1%	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in x around 0 71.3%

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

   4. Simplified 71.2%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
      Step-by-step derivation

      [Start] 71.3%	\[ F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{{F}^{2} + 2}}\right) - \frac{x}{\tan B} \]
      associate-*l/ [=>] 71.2%	\[ F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{{F}^{2} + 2}}}{\sin B}} - \frac{x}{\tan B} \]
      *-lft-identity [=>] 71.2%	\[ F \cdot \frac{\color{blue}{\sqrt{\frac{1}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [=>] 71.2%	\[ F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]
      fma-udef [<=] 71.2%	\[ F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   5. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	26696

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -68000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 500:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Accuracy	98.9%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5200000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Accuracy	90.8%
Cost	14344

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -310:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 6
Accuracy	90.6%
Cost	14152

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5200000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-58}:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Accuracy	90.6%
Cost	14024

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5200000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 8
Accuracy	82.7%
Cost	13644

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-233}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 9
Accuracy	90.6%
Cost	13640

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5200000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 10
Accuracy	68.9%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -108000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-233}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 11
Accuracy	68.9%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -5200000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-233}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 12
Accuracy	75.6%
Cost	13448

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-233}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 13
Accuracy	63.3%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;F \leq -6000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-292}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 14
Accuracy	62.3%
Cost	7245

\[\begin{array}{l} \mathbf{if}\;F \leq -23500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-302} \lor \neg \left(F \leq 5.5 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \end{array} \]

Alternative 15
Accuracy	54.9%
Cost	7244

\[\begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;B \leq -2.1 \cdot 10^{-195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-263}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Accuracy	52.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;F \leq 10^{-294} \lor \neg \left(F \leq 7.5 \cdot 10^{-115}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \end{array} \]

Alternative 17
Accuracy	43.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -2.85 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 18
Accuracy	43.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.46 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-113}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 19
Accuracy	36.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq 1.55 \cdot 10^{-113}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20
Accuracy	29.5%
Cost	256

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

Reproduce

herbie shell --seed 2023167 
(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))))))