VandenBroeck and Keller, Equation (23)

Average Accuracy: 78.3% → 99.6%

Time: 33.9s

Precision: binary64

Cost: 33288

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{-\tan B}{\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}}{\frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

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
 (if (<= F -5e+31)
   (/ (- (/ (- (tan B)) (sin B)) x) (tan B))
   (if (<= F 2.5e+24)
     (-
      (/ (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (/ 1.0 F))
      (/ x (tan B)))
     (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B))))))

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 tmp;
	if (F <= -5e+31) {
		tmp = ((-tan(B) / sin(B)) - x) / tan(B);
	} else if (F <= 2.5e+24) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)) / (1.0 / F)) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	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)
	tmp = 0.0
	if (F <= -5e+31)
		tmp = Float64(Float64(Float64(Float64(-tan(B)) / sin(B)) - x) / tan(B));
	elseif (F <= 2.5e+24)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)) / Float64(1.0 / F)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	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_] := If[LessEqual[F, -5e+31], N[(N[(N[((-N[Tan[B], $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e+24], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000027e31

   1. Initial program 58.6%

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

   2. Simplified 58.7%

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

      [Start] 58.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 58.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 58.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 58.6	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 58.6	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 58.6	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 58.6	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 58.6	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 58.6	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 58.7	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 58.7	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Taylor expanded in F around -inf 99.7%

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

   4. Applied egg-rr 80.7%

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

   5. Simplified 99.7%

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

      [Start] 80.7	\[ \left(\left(-\tan B\right) - \sin B \cdot x\right) \cdot \frac{\frac{1}{\sin B}}{\tan B} \]
      associate-*r/ [=>] 99.6	\[ \color{blue}{\frac{\left(\left(-\tan B\right) - \sin B \cdot x\right) \cdot \frac{1}{\sin B}}{\tan B}} \]
      associate-*r/ [=>] 99.7	\[ \frac{\color{blue}{\frac{\left(\left(-\tan B\right) - \sin B \cdot x\right) \cdot 1}{\sin B}}}{\tan B} \]
      associate-*l/ [<=] 99.7	\[ \frac{\color{blue}{\frac{\left(-\tan B\right) - \sin B \cdot x}{\sin B} \cdot 1}}{\tan B} \]
      associate-/l* [=>] 99.7	\[ \color{blue}{\frac{\frac{\left(-\tan B\right) - \sin B \cdot x}{\sin B}}{\frac{\tan B}{1}}} \]
      div-sub [=>] 99.7	\[ \frac{\color{blue}{\frac{-\tan B}{\sin B} - \frac{\sin B \cdot x}{\sin B}}}{\frac{\tan B}{1}} \]
      *-commutative [=>] 99.7	\[ \frac{\frac{-\tan B}{\sin B} - \frac{\color{blue}{x \cdot \sin B}}{\sin B}}{\frac{\tan B}{1}} \]
      associate-/l* [=>] 99.7	\[ \frac{\frac{-\tan B}{\sin B} - \color{blue}{\frac{x}{\frac{\sin B}{\sin B}}}}{\frac{\tan B}{1}} \]
      *-inverses [=>] 99.7	\[ \frac{\frac{-\tan B}{\sin B} - \frac{x}{\color{blue}{1}}}{\frac{\tan B}{1}} \]
      /-rgt-identity [=>] 99.7	\[ \frac{\frac{-\tan B}{\sin B} - \frac{x}{1}}{\color{blue}{\tan B}} \]

   if -5.00000000000000027e31 < F < 2.50000000000000023e24

   1. Initial program 99.1%

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

   2. Simplified 99.2%

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

      [Start] 99.1	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 99.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 99.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 99.1	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 99.1	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 99.1	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 99.1	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 99.1	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 99.1	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 99.2	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 99.2	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}}{\frac{1}{F}}} - \frac{x}{\tan B} \]

   if 2.50000000000000023e24 < F

   1. Initial program 57.1%

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

   2. Taylor expanded in x around 0 57.1%

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

   3. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{\cos B \cdot x}{\sin B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{-\tan B}{\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}}{\frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	33160

\[\begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{-\tan B}{\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+24}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	27144

\[\begin{array}{l} \mathbf{if}\;F \leq -3.65 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{-\tan B}{\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 90000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	20744

\[\begin{array}{l} \mathbf{if}\;F \leq -3.65 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{-\tan B}{\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 32000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	20744

\[\begin{array}{l} \mathbf{if}\;F \leq -3.65 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{-\tan B}{\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 180000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Alternative 5
Accuracy	99.0%
Cost	20552

\[\begin{array}{l} \mathbf{if}\;F \leq -1.65:\\ \;\;\;\;\frac{\frac{-\tan B}{\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B}}{\frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Accuracy	99.1%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.48:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Alternative 8
Accuracy	99.0%
Cost	20040

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

Alternative 9
Accuracy	91.1%
Cost	14476

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.13:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-101}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{elif}\;F \leq 500000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 10
Accuracy	91.1%
Cost	14348

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.37:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{elif}\;F \leq 1800000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 11
Accuracy	90.6%
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{x}{\tan B}\\ t_2 := \frac{F}{B} \cdot t_0 - t_1\\ \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.355:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 12
Accuracy	90.7%
Cost	14024

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.68:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.215:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 13
Accuracy	82.8%
Cost	13776

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6000:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-55}:\\ \;\;\;\;t_0 \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-185}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 170000:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 14
Accuracy	67.5%
Cost	13580

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;F \leq -3.65 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;t_0 \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-182}:\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{elif}\;F \leq 220000:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 15
Accuracy	67.5%
Cost	13580

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;F \leq -3.65 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-50}:\\ \;\;\;\;t_0 \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 230000:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 16
Accuracy	75.7%
Cost	13580

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;F \leq -6000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;t_0 \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-182}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17
Accuracy	58.9%
Cost	8200

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.65 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-55}:\\ \;\;\;\;t_0 \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 170000:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 18
Accuracy	58.9%
Cost	8080

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.65 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 19
Accuracy	59.2%
Cost	7760

\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.045:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -4.15 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 0.65:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 20
Accuracy	48.7%
Cost	7641

\[\begin{array}{l} t_0 := \frac{1 - x}{B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-229}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-110} \lor \neg \left(x \leq 3.2 \cdot 10^{-43}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 21
Accuracy	55.2%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-210}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 22
Accuracy	41.3%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -14600:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 23
Accuracy	37.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{-59}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 24
Accuracy	29.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 25
Accuracy	25.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;F \leq -6800:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 26
Accuracy	11.1%
Cost	192

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

Reproduce

herbie shell --seed 2023122 
(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))))))