VandenBroeck and Keller, Equation (23)

Average Error: 14.1 → 0.2

Time: 30.2s

Precision: binary64

Cost: 33160

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1060000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

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 -1060000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100000000.0)
       (- (* (/ F (sin B)) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) 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 <= -1060000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = ((F / sin(B)) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - 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 <= -1060000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - 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, -1060000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -1.06e9

   1. Initial program 26.5

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

   2. Simplified 26.4

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

      [Start] 26.5	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 26.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 26.5	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 26.5	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 26.5	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 26.5	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 26.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 26.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 26.5	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 26.4	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 26.4	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Taylor expanded in F around -inf 0.1

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

   if -1.06e9 < F < 1e8

   1. Initial program 0.4

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

   2. Simplified 0.3

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

      [Start] 0.4	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 0.4	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 0.4	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 0.4	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 0.4	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 0.4	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 0.4	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 0.4	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 0.4	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 0.3	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 0.3	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   if 1e8 < F

   1. Initial program 25.0

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

   2. Simplified 25.0

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

      [Start] 25.0	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 25.0	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 25.0	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 25.0	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Taylor expanded in F around inf 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1060000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	33096

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Error	0.5
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{+82}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 95000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	20424

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2.8:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Error	0.7
Cost	20040

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

Alternative 5
Error	7.5
Cost	14480

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1060000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -9 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2500000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 6
Error	8.1
Cost	14288

\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0062:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.48:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 7
Error	7.9
Cost	14288

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.08:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.115:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 8
Error	21.1
Cost	13580

\[\begin{array}{l} t_0 := 2 + F \cdot F\\ \mathbf{if}\;F \leq -3400000000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.4 \cdot 10^{-124}:\\ \;\;\;\;{\left(t_0 + x \cdot 2\right)}^{-0.5} \cdot \left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_0}} - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 9
Error	10.6
Cost	13512

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 10
Error	15.8
Cost	13380

\[\begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 11
Error	21.1
Cost	8200

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := 2 + F \cdot F\\ \mathbf{if}\;F \leq -3400000000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-128}:\\ \;\;\;\;{\left(t_1 + x \cdot 2\right)}^{-0.5} \cdot \left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_1}} - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 12
Error	21.3
Cost	8008

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(F \cdot \left(B \cdot 0.16666666666666666 + \frac{1}{B}\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13
Error	21.3
Cost	7764

\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}} - x}{B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -9000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14
Error	23.0
Cost	7500

\[\begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{+17}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + F \cdot F}}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15
Error	23.1
Cost	7500

\[\begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 16
Error	21.0
Cost	7368

\[\begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17
Error	26.5
Cost	7184

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Error	26.5
Cost	7184

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+232}:\\ \;\;\;\;-1 + \left(t_1 + 1\right)\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19
Error	23.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 20
Error	34.3
Cost	6920

\[\begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 21
Error	35.1
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 22
Error	37.7
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 23
Error	40.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;F \leq -300:\\ \;\;\;\;B \cdot -0.16666666666666666 + \left(\frac{-1}{B} - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 24
Error	40.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 25
Error	45.1
Cost	520

\[\begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 26
Error	42.4
Cost	520

\[\begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 27
Error	51.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;F \leq 7.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 28
Error	56.8
Cost	192

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

Reproduce

herbie shell --seed 2023059 
(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))))))