VandenBroeck and Keller, Equation (23)

Average Error: 78.7% → 99.3%

Time: 44.2s

Precision: binary64

Cost: 26568.00

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

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+156)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.023)
       (- (/ (/ F (sqrt (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 <= -2e+156) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.023) {
		tmp = ((F / sqrt(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 <= -2e+156)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.023)
		tmp = Float64(Float64(Float64(F / sqrt(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, -2e+156], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.023], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 < -2e156

   1. Initial program 34.1

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

   2. Simplified 34.2

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

      [Start] 34.1	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 34.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 34.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 34.1	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 34.1	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 34.1	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 34.1	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 34.1	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 34.1	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 34.2	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 34.2	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Taylor expanded in F around -inf 99.8

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

   if -2e156 < F < 0.023

   1. Initial program 97.5

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

   2. Simplified 97.7

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

      [Start] 97.5	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 97.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 97.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 97.5	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 97.5	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 97.5	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 97.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 97.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 97.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 97.7	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 97.7	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Applied egg-rr 97.7

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

   4. Applied egg-rr 99.5

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

   5. Taylor expanded in x around 0 99.4

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

   6. Simplified 99.4

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

      [Start] 99.4	\[ \frac{F \cdot \sqrt{\frac{1}{{F}^{2} + 2}}}{\sin B} - \frac{x}{\tan B} \]
      unpow2 [=>] 99.4	\[ \frac{F \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]
      fma-udef [<=] 99.4	\[ \frac{F \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Applied egg-rr 72.4

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}\right)} - 1}}{\sin B} - \frac{x}{\tan B} \]

   8. Simplified 99.5

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

      [Start] 72.4	\[ \frac{e^{\mathsf{log1p}\left(\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}\right)} - 1}{\sin B} - \frac{x}{\tan B} \]
      expm1-def [=>] 99.5	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}\right)\right)}}{\sin B} - \frac{x}{\tan B} \]
      expm1-log1p [=>] 99.5	\[ \frac{\color{blue}{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   if 0.023 < F

   1. Initial program 60.5

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

   2. Simplified 60.5

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

      [Start] 60.5	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 60.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 60.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 60.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Taylor expanded in F around inf 98.8

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Error	99.3%
Cost	20744.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Error	99.1%
Cost	20484.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.3:\\ \;\;\;\;\frac{-1 + \left(\frac{1}{F \cdot F} + \frac{-1.5}{{F}^{4}}\right)}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Error	99.0%
Cost	20424.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Error	99.0%
Cost	20040.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Error	91.1%
Cost	14284.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{if}\;F \leq -0.39:\\ \;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_1 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + t_1 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 6
Error	91.2%
Cost	14284.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -245000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -1.16 \cdot 10^{-82}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Error	91.1%
Cost	14024.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.98:\\ \;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 8
Error	88.1%
Cost	13904.00

\[\begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.28:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{elif}\;F \leq 0.0108:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 9
Error	91.0%
Cost	13900.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.29:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 10
Error	91.1%
Cost	13900.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.8:\\ \;\;\;\;\frac{-1 + \frac{1}{F \cdot F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 11
Error	65.2%
Cost	13712.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -7 \cdot 10^{+271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 10^{-58}:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 12
Error	65.2%
Cost	13712.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 13
Error	82.3%
Cost	13512.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 14
Error	74.2%
Cost	13448.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 15
Error	51.3%
Cost	7508.00

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.22 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-146}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	55.5%
Cost	7244.00

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]

Alternative 17
Error	45.1%
Cost	7048.00

\[\begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.16 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Error	45.5%
Cost	6856.00

\[\begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 95000000:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19
Error	41.3%
Cost	6724.00

\[\begin{array}{l} \mathbf{if}\;F \leq -5.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20
Error	36.6%
Cost	840.00

\[\begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-42}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Error	36.7%
Cost	584.00

\[\begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 22
Error	28.9%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 23
Error	25.1%
Cost	388.00

\[\begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 24
Error	10.9%
Cost	192.00

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

Reproduce

herbie shell --seed 2023093 
(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))))))