VandenBroeck and Keller, Equation (23)

Average Accuracy: 78.4% → 99.6%

Time: 31.6s

Precision: binary64

Cost: 26696

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;F \cdot \frac{t_0}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_1\\ \end{array} \]

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 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -1.65e+85)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 40000000.0)
       (- (* F (/ t_0 (sqrt (fma F F 2.0)))) t_1)
       (- t_0 t_1)))))

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 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.65e+85) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 40000000.0) {
		tmp = (F * (t_0 / sqrt(fma(F, F, 2.0)))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	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(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+85)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 40000000.0)
		tmp = Float64(Float64(F * Float64(t_0 / sqrt(fma(F, F, 2.0)))) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+85], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 40000000.0], N[(N[(F * N[(t$95$0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e85

   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 59.5%

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

      [Start] 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/ [=>] 59.5	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 59.4	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 59.4	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 99.8%

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

   if -1.65e85 < F < 4e7

   1. Initial program 98.6%

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

   2. Simplified 99.5%

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

      [Start] 98.6	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 98.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 98.6	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 99.3	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 99.3	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in x around 0 99.5%

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

   4. Simplified 99.5%

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

      [Start] 99.5	\[ F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{{F}^{2} + 2}}\right) - \frac{x}{\tan B} \]
      *-commutative [=>] 99.5	\[ F \cdot \color{blue}{\left(\sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{1}{\sin B}\right)} - \frac{x}{\tan B} \]
      unpow2 [=>] 99.5	\[ F \cdot \left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B} \]
      fma-udef [<=] 99.5	\[ F \cdot \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B} \]

   5. Applied egg-rr 99.6%

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

      [Start] 99.5	\[ F \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B} \]
      *-commutative [=>] 99.5	\[ F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\right)} - \frac{x}{\tan B} \]
      sqrt-div [=>] 99.5	\[ F \cdot \left(\frac{1}{\sin B} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}\right) - \frac{x}{\tan B} \]
      metadata-eval [=>] 99.5	\[ F \cdot \left(\frac{1}{\sin B} \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}\right) - \frac{x}{\tan B} \]
      un-div-inv [=>] 99.6	\[ F \cdot \color{blue}{\frac{\frac{1}{\sin B}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}} - \frac{x}{\tan B} \]

   if 4e7 < F

   1. Initial program 58.2%

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

   2. Simplified 75.0%

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

      [Start] 58.2	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 58.2	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 58.2	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 75.1	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 74.9	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 74.9	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;F \cdot \frac{\frac{1}{\sin B}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	26568

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1550000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -0.91:\\ \;\;\;\;F \cdot \frac{t_1}{\frac{-1}{F} - F} - t_0\\ \mathbf{elif}\;F \leq 0.92:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{t_1}{F + \frac{1}{F}} - t_0\\ \end{array} \]

Alternative 4
Accuracy	88.3%
Cost	14480

\[\begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := t_0 \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1550000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_0}{\frac{-1}{F} - F} - t_2\\ \mathbf{elif}\;F \leq 23000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_2\\ \end{array} \]

Alternative 5
Accuracy	85.4%
Cost	14288

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-81}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\frac{-1}{F} - F} - t_0\\ \mathbf{elif}\;F \leq 0.00024:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{\sin B}}{F + \frac{1}{F}} - t_0\\ \end{array} \]

Alternative 6
Accuracy	85.5%
Cost	14288

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{-1}{F} - F\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{-16}:\\ \;\;\;\;F \cdot \frac{t_1}{t_2} - t_0\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-78}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{t_2} - t_0\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{t_1}{F + \frac{1}{F}} - t_0\\ \end{array} \]

Alternative 7
Accuracy	85.3%
Cost	14028

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\frac{-1}{F} - F} - t_0\\ \mathbf{elif}\;F \leq 0.016:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 8
Accuracy	66.5%
Cost	13912

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := F \cdot \frac{\sqrt{0.5}}{\sin B}\\ t_2 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4.8 \cdot 10^{+199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -5.3 \cdot 10^{+182}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;1 + \left(-1 + t_0\right)\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 0.0035:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{B}\\ \end{array} \]

Alternative 9
Accuracy	66.4%
Cost	13912

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4.1 \cdot 10^{+199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.95 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;1 + \left(-1 + t_0\right)\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 0.018:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{B}\\ \end{array} \]

Alternative 10
Accuracy	85.4%
Cost	13776

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-80}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 10^{-84}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.02:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 11
Accuracy	77.0%
Cost	13648

\[\begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.1 \cdot 10^{-77}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 4.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{B}\\ \end{array} \]

Alternative 12
Accuracy	64.6%
Cost	8024

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_2 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.15 \cdot 10^{+201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3 \cdot 10^{+183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq -86000:\\ \;\;\;\;1 + \left(-1 + t_0\right)\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{B}\\ \end{array} \]

Alternative 13
Accuracy	60.3%
Cost	7582

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -86000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-26} \lor \neg \left(F \leq 7.2 \cdot 10^{+194} \lor \neg \left(F \leq 9 \cdot 10^{+246}\right) \land F \leq 1.85 \cdot 10^{+289}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 14
Accuracy	60.1%
Cost	7582

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{+199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -95000:\\ \;\;\;\;1 + \left(-1 + t_0\right)\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-26} \lor \neg \left(F \leq 6 \cdot 10^{+194} \lor \neg \left(F \leq 5.5 \cdot 10^{+252}\right) \land F \leq 2.2 \cdot 10^{+287}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15
Accuracy	63.5%
Cost	7504

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.9 \cdot 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -90000:\\ \;\;\;\;1 + \left(-1 + t_0\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 16
Accuracy	63.4%
Cost	7504

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4.8 \cdot 10^{+199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7.7 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -130000000:\\ \;\;\;\;1 + \left(-1 + t_0\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{B}\\ \end{array} \]

Alternative 17
Accuracy	47.5%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;F \leq -86000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Accuracy	46.1%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -86000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19
Accuracy	41.0%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -86000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20
Accuracy	33.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -86000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Accuracy	33.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{+21}:\\ \;\;\;\;B \cdot -0.16666666666666666 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 22
Accuracy	30.3%
Cost	520

\[\begin{array}{l} \mathbf{if}\;F \leq -160000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 23
Accuracy	19.4%
Cost	324

\[\begin{array}{l} \mathbf{if}\;F \leq 8.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 24
Accuracy	11.8%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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))))))