VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.4%

Time: 25.4s

Precision: binary64

Cost: 33160

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

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 -2e+113)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.00039)
       (- (/ (* F (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (sin B)) t_0)
       (- (/ (+ 1.0 (/ (- -1.0 x) (* F F))) (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 <= -2e+113) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.00039) {
		tmp = ((F * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) / sin(B)) - t_0;
	} else {
		tmp = ((1.0 + ((-1.0 - x) / (F * F))) / 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 <= -2e+113)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.00039)
		tmp = Float64(Float64(Float64(F * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-1.0 - x) / Float64(F * F))) / 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, -2e+113], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.00039], N[(N[(N[(F * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e113

   1. Initial program 46.0%

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

   2. Simplified 61.5%

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

      [Start] 46.0	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 46.0	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 46.0	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 61.5	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 61.6	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 61.6	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 99.8%

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

   if -2e113 < F < 3.89999999999999993e-4

   1. Initial program 98.7%

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

   2. Simplified 99.7%

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

      [Start] 98.7	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 98.7	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 98.7	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      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/ [<=] 99.4	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-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. Applied egg-rr 99.8%

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

      [Start] 99.7	\[ F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]
      associate-*r/ [=>] 99.8	\[ \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   if 3.89999999999999993e-4 < F

   1. Initial program 57.5%

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

   2. Simplified 77.6%

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

      [Start] 57.5	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 57.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 57.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 77.6	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 77.5	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 77.5	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Applied egg-rr 77.7%

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

      [Start] 77.6	\[ F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]
      associate-*r/ [=>] 77.7	\[ \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   4. Taylor expanded in F around inf 99.8%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot \frac{2 + 2 \cdot x}{{F}^{2}}}}{\sin B} - \frac{x}{\tan B} \]

   5. Simplified 99.8%

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

      [Start] 99.8	\[ \frac{1 + -0.5 \cdot \frac{2 + 2 \cdot x}{{F}^{2}}}{\sin B} - \frac{x}{\tan B} \]
      associate-*r/ [=>] 99.8	\[ \frac{1 + \color{blue}{\frac{-0.5 \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}}}{\sin B} - \frac{x}{\tan B} \]
      distribute-lft-in [=>] 99.8	\[ \frac{1 + \frac{\color{blue}{-0.5 \cdot 2 + -0.5 \cdot \left(2 \cdot x\right)}}{{F}^{2}}}{\sin B} - \frac{x}{\tan B} \]
      metadata-eval [=>] 99.8	\[ \frac{1 + \frac{\color{blue}{-1} + -0.5 \cdot \left(2 \cdot x\right)}{{F}^{2}}}{\sin B} - \frac{x}{\tan B} \]
      associate-*r* [=>] 99.8	\[ \frac{1 + \frac{-1 + \color{blue}{\left(-0.5 \cdot 2\right) \cdot x}}{{F}^{2}}}{\sin B} - \frac{x}{\tan B} \]
      metadata-eval [=>] 99.8	\[ \frac{1 + \frac{-1 + \color{blue}{-1} \cdot x}{{F}^{2}}}{\sin B} - \frac{x}{\tan B} \]
      mul-1-neg [=>] 99.8	\[ \frac{1 + \frac{-1 + \color{blue}{\left(-x\right)}}{{F}^{2}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [=>] 99.8	\[ \frac{1 + \frac{-1 + \left(-x\right)}{\color{blue}{F \cdot F}}}{\sin B} - \frac{x}{\tan B} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	33160

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -95000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	33160

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	27144

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -215000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -215000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 8
Accuracy	88.3%
Cost	14288

\[\begin{array}{l} t_0 := F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-210}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 9
Accuracy	91.5%
Cost	14024

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.025:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 10
Accuracy	91.5%
Cost	14024

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.031:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.00039:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1 - x}{F \cdot F}}{\sin B} - t_0\\ \end{array} \]

Alternative 11
Accuracy	84.0%
Cost	13512

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-45}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 12
Accuracy	71.0%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -0.031:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.22 \cdot 10^{-13}:\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+83} \lor \neg \left(F \leq 3.8 \cdot 10^{+230}\right):\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 13
Accuracy	71.0%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -0.00165:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.22 \cdot 10^{-13}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{+83} \lor \neg \left(F \leq 1.12 \cdot 10^{+231}\right):\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 14
Accuracy	71.0%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -0.033:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.22 \cdot 10^{-13}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+83} \lor \neg \left(F \leq 1.8 \cdot 10^{+231}\right):\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 15
Accuracy	77.9%
Cost	13448

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2.22 \cdot 10^{-13}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{+83} \lor \neg \left(F \leq 6.9 \cdot 10^{+230}\right):\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 16
Accuracy	60.6%
Cost	7897

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.033:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-202}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-233}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+31} \lor \neg \left(F \leq 3.8 \cdot 10^{+83}\right) \land F \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17
Accuracy	62.9%
Cost	7765

\[\begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+32} \lor \neg \left(F \leq 3.8 \cdot 10^{+83}\right) \land F \leq 3 \cdot 10^{+230}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 18
Accuracy	59.9%
Cost	7509

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.033:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(-F\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-208}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-233}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+39} \lor \neg \left(F \leq 9.4 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19
Accuracy	55.2%
Cost	7508

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Accuracy	44.6%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -215000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 21
Accuracy	43.9%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -215000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 22
Accuracy	42.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -215000:\\ \;\;\;\;-1 + \left(1 + \frac{-1 - x}{B}\right)\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 23
Accuracy	42.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -215000:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 24
Accuracy	42.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;F \leq -215000:\\ \;\;\;\;-1 + \left(1 + \frac{-1 - x}{B}\right)\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 25
Accuracy	42.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 26
Accuracy	35.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 27
Accuracy	28.4%
Cost	256

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

Alternative 28
Accuracy	10.6%
Cost	192

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

Reproduce

herbie shell --seed 2023161 
(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))))))