VandenBroeck and Keller, Equation (23)

Average Accuracy: 77.9% → 99.6%

Time: 43.2s

Precision: binary64

Cost: 33160

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 10^{+55}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

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
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4e+21)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1e+55)
       (- (/ F (/ (sin B) (pow (fma x 2.0 (fma F F 2.0)) -0.5))) t_0)
       (+ (/ 1.0 (sin B)) (* x (/ -1.0 (tan 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 t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4e+21) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1e+55) {
		tmp = (F / (sin(B) / pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(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)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4e+21)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1e+55)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(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_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4e+21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e+55], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4e21

   1. Initial program 58.3%

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

   2. Simplified 68.3%

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

      [Start] 58.3	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 58.3	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 58.3	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 68.3	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 68.2	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 68.2	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 99.8%

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

   if -1.4e21 < F < 1.00000000000000001e55

   1. Initial program 99.2%

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

   2. Simplified 99.6%

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

      [Start] 99.2	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 99.2	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 99.2	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      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. Applied egg-rr 99.6%

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

      [Start] 99.6	\[ F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]
      clear-num [=>] 99.5	\[ F \cdot \color{blue}{\frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]
      un-div-inv [=>] 99.6	\[ \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]

   if 1.00000000000000001e55 < F

   1. Initial program 52.6%

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

   2. Simplified 63.5%

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

      [Start] 52.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 52.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 52.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 63.4	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 63.4	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 63.4	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around inf 99.8%

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

   4. Applied egg-rr 99.7%

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

      [Start] 99.8	\[ \frac{1}{\sin B} - \frac{x}{\tan B} \]
      clear-num [=>] 99.7	\[ \frac{1}{\sin B} - \color{blue}{\frac{1}{\frac{\tan B}{x}}} \]
      associate-/r/ [=>] 99.7	\[ \frac{1}{\sin B} - \color{blue}{\frac{1}{\tan B} \cdot x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+55}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	33160

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -105000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 4 \cdot 10^{+56}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Accuracy	99.0%
Cost	20424

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{\frac{1}{2 + x \cdot 2}}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Accuracy	91.4%
Cost	14028

\[\begin{array}{l} t_0 := 2 + x \cdot 2\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.42:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{\frac{1}{t_0}}}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.074:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{t_0}} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 6
Accuracy	91.4%
Cost	14028

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -8.3 \cdot 10^{-100}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + F \cdot F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.09:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Accuracy	90.8%
Cost	13896

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.42:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.082:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 8
Accuracy	82.7%
Cost	13512

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.16 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 9
Accuracy	67.7%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 10
Accuracy	67.8%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq -1.72 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 11
Accuracy	75.2%
Cost	13448

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 12
Accuracy	59.4%
Cost	7944

\[\begin{array}{l} \mathbf{if}\;F \leq -22000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9 \cdot 10^{-272}:\\ \;\;\;\;{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+72} \lor \neg \left(F \leq 6.9 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13
Accuracy	59.4%
Cost	7816

\[\begin{array}{l} \mathbf{if}\;F \leq -21500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-272}:\\ \;\;\;\;{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+70} \lor \neg \left(F \leq 5.8 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14
Accuracy	52.0%
Cost	7509

\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.22 \cdot 10^{-220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.75 \cdot 10^{-272}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+26} \lor \neg \left(F \leq 5.3 \cdot 10^{+223}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15
Accuracy	55.8%
Cost	7509

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+96}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -1.68 \cdot 10^{-270}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{+70} \lor \neg \left(F \leq 1.6 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 16
Accuracy	56.7%
Cost	7509

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.85 \cdot 10^{+95}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -4.7 \cdot 10^{-271}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+71} \lor \neg \left(F \leq 4.5 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17
Accuracy	52.4%
Cost	7377

\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.4 \cdot 10^{-221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-271}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+26} \lor \neg \left(F \leq 5.3 \cdot 10^{+223}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Accuracy	45.8%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -1.72 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19
Accuracy	41.8%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-40}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20
Accuracy	38.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Accuracy	26.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;1 + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 22
Accuracy	30.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 23
Accuracy	26.2%
Cost	388

\[\begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 24
Accuracy	11.1%
Cost	192

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

Reproduce

herbie shell --seed 2023144 
(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))))))