VandenBroeck and Keller, Equation (23)

Average Accuracy: 78.3% → 99.6%

Time: 36.6s

Precision: binary64

Cost: 33160

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

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 -5e+75)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 17000.0)
       (- (/ (* F (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (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 <= -5e+75) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 17000.0) {
		tmp = ((F * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) / 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 <= -5e+75)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 17000.0)
		tmp = Float64(Float64(Float64(F * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) / 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, -5e+75], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 17000.0], N[(N[(N[(F * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -5.0000000000000002e75

   1. Initial program 53.2%

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

   2. Simplified 62.6%

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

      [Start] 53.2	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 53.2	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 53.2	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 62.5	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 62.5	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 62.5	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around -inf 99.7%

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

   if -5.0000000000000002e75 < F < 17000

   1. Initial program 98.9%

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

   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] 98.9	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 98.9	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 98.9	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      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%

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

      [Start] 99.5	\[ F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]
      associate-*r/ [=>] 99.6	\[ \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   if 17000 < F

   1. Initial program 58.1%

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

   2. Simplified 67.9%

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

      [Start] 58.1	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 58.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 58.1	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 67.9	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 67.8	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 67.8	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around inf 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	26696

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	26568

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -150000:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -200000:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -890:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	20040

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -890:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 7
Accuracy	84.2%
Cost	14672

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + F \cdot F}} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.16 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{\frac{\sin B}{-\cos B}}\\ \mathbf{elif}\;F \leq 3400:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 8
Accuracy	87.7%
Cost	14480

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00095:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{\frac{\sin B}{-\cos B}}\\ \mathbf{elif}\;F \leq 9500:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 9
Accuracy	83.1%
Cost	13776

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -9 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\frac{\sin B}{-\cos B}}\\ \mathbf{elif}\;F \leq 115:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 10
Accuracy	83.3%
Cost	13776

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\frac{\sin B}{-\cos B}}\\ \mathbf{elif}\;F \leq 4.5:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 11
Accuracy	65.1%
Cost	13580

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ t_1 := F \cdot \sin B\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00095:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - t_2\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\frac{\sin B}{-\cos B}}\\ \mathbf{elif}\;F \leq 0.038:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 17500:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+116}:\\ \;\;\;\;F \cdot \frac{1}{t_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{B} - t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{t_1} - \frac{x}{B}\\ \end{array} \]

Alternative 12
Accuracy	76.3%
Cost	13580

\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ t_1 := F \cdot \sin B\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\frac{\sin B}{-\cos B}}\\ \mathbf{elif}\;F \leq 10000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 17500:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;F \cdot \frac{1}{t_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.15 \cdot 10^{+239}:\\ \;\;\;\;\frac{1}{B} - t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{t_1} - \frac{x}{B}\\ \end{array} \]

Alternative 13
Accuracy	65.2%
Cost	8464

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{x}{\tan B}\\ t_2 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ t_3 := F \cdot \sin B\\ \mathbf{if}\;F \leq -0.00095:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - t_1\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 18500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+117}:\\ \;\;\;\;F \cdot \frac{1}{t_3} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{B} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{t_3} - \frac{x}{B}\\ \end{array} \]

Alternative 14
Accuracy	63.3%
Cost	7769

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -9.4 \cdot 10^{+256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{+134}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 17500:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+116} \lor \neg \left(F \leq 1.16 \cdot 10^{+231}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 15
Accuracy	63.3%
Cost	7768

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{+256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{+135}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 17500:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{+117}:\\ \;\;\;\;F \cdot \frac{1}{t_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+233}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{t_0} - \frac{x}{B}\\ \end{array} \]

Alternative 16
Accuracy	64.1%
Cost	7504

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1300:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - t_1\\ \mathbf{elif}\;F \leq 17500:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{+116}:\\ \;\;\;\;F \cdot \frac{1}{t_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;\frac{1}{B} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{t_0} - \frac{x}{B}\\ \end{array} \]

Alternative 17
Accuracy	57.1%
Cost	7449

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{+256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{+135}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{+27} \lor \neg \left(F \leq 10^{+117}\right) \land F \leq 1.22 \cdot 10^{+231}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Accuracy	42.3%
Cost	7385

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -1.6 \cdot 10^{+258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{+132}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{+117} \lor \neg \left(F \leq 8.4 \cdot 10^{+179}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \end{array} \]

Alternative 19
Accuracy	62.4%
Cost	7376

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.42 \cdot 10^{+135}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 20
Accuracy	39.7%
Cost	6988

\[\begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{+256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+132}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-42}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Accuracy	37.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 22
Accuracy	37.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -1.46 \cdot 10^{-18}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 - \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 23
Accuracy	22.8%
Cost	786

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-175} \lor \neg \left(x \leq 9.5 \cdot 10^{-228} \lor \neg \left(x \leq 6 \cdot 10^{-89}\right) \land x \leq 2.3 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]

Alternative 24
Accuracy	37.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 25
Accuracy	29.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -1.48 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 26
Accuracy	10.7%
Cost	192

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

Reproduce

herbie shell --seed 2023146 
(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))))))