VandenBroeck and Keller, Equation (23)

Average Error: 22.72% → 0.37%

Time: 27.0s

Precision: binary64

Cost: 33160

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - t_0\\ \end{array} \]

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 -1e+15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2e+97)
       (- (/ (* F (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (sin B)) t_0)
       (- (/ F (* F (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 <= -1e+15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2e+97) {
		tmp = ((F * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) / sin(B)) - t_0;
	} else {
		tmp = (F / (F * 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 <= -1e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2e+97)
		tmp = Float64(Float64(Float64(F * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(F / Float64(F * 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, -1e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2e+97], 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[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e15

   1. Initial program 41.05

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

   2. Simplified 40.99

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

      [Start] 41.05	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 41.05	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 41.05	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 41.05	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 41.05	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 41.05	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 41.05	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 41.05	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 41.05	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 40.99	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 40.99	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Taylor expanded in F around -inf 0.23

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

   if -1e15 < F < 2.0000000000000001e97

   1. Initial program 1.41

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

   2. Simplified 1.28

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

      [Start] 1.41	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 1.41	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 1.41	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 1.41	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 1.41	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 1.41	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 1.41	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 1.41	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 1.41	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 1.28	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 1.28	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Applied egg-rr 0.44

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

   if 2.0000000000000001e97 < F

   1. Initial program 53.68

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

   2. Simplified 53.55

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

      [Start] 53.68	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 53.68	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 53.68	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      +-commutative [=>] 53.68	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      *-commutative [=>] 53.68	\[ \frac{F}{\sin B} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 53.62	\[ \frac{F}{\sin B} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      fma-def [=>] 53.62	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 53.62	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)} - x \cdot \frac{1}{\tan B} \]
      metadata-eval [=>] 53.62	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [=>] 53.55	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \color{blue}{\frac{x \cdot 1}{\tan B}} \]
      *-rgt-identity [=>] 53.55	\[ \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{\color{blue}{x}}{\tan B} \]

   3. Applied egg-rr 42.75

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

   4. Taylor expanded in F around inf 0.38

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

4. Final simplification 0.37

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.38%
Cost	33160

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 10^{+75}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - t_0\\ \end{array} \]

Alternative 2
Error	0.38%
Cost	26696

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 135000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\frac{\sin B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 3
Error	0.41%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1000000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 4
Error	0.41%
Cost	20744

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 5
Error	1.08%
Cost	20040

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

Alternative 6
Error	1.08%
Cost	20040

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

Alternative 7
Error	11.2%
Cost	14480

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -2.65 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 640:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 8
Error	11.64%
Cost	14288

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-228}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.0145:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]

Alternative 9
Error	33.13%
Cost	13712

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{F}{F \cdot \left(-\sin B\right)} - \frac{x}{B}\\ \mathbf{if}\;F \leq -7.2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -8 \cdot 10^{+69}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 16:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{+136} \lor \neg \left(F \leq 2.75 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 10
Error	33.04%
Cost	13712

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{F}{F \cdot \left(-\sin B\right)} - \frac{x}{B}\\ \mathbf{if}\;F \leq -8.2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{+68}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 20:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{+137} \lor \neg \left(F \leq 2.35 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 11
Error	15.99%
Cost	13644

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 11.5:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]

Alternative 12
Error	24.92%
Cost	13448

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 20:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.62 \cdot 10^{+140} \lor \neg \left(F \leq 2.75 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 13
Error	40.66%
Cost	8344

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t_0\\ t_2 := \frac{F}{F \cdot \left(-\sin B\right)} - \frac{x}{B}\\ t_3 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.55 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq -1.26 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -4.4:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-134}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq -4.7 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 490:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq 1.28 \cdot 10^{+137} \lor \neg \left(F \leq 2.6 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 14
Error	44.41%
Cost	7765

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{F}{F \cdot \left(-\sin B\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-215}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-163}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+139} \lor \neg \left(F \leq 6.5 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 15
Error	44.6%
Cost	7641

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t_0\\ \mathbf{if}\;F \leq -5 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-163}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+136} \lor \neg \left(F \leq 7 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 16
Error	44.14%
Cost	7509

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.25 \cdot 10^{+122}:\\ \;\;\;\;\frac{F}{F \cdot \left(-\sin B\right)} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-220}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-163}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+139} \lor \neg \left(F \leq 1.22 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 17
Error	47.01%
Cost	7508

\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{+175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{+122}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-211}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-158}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;F \leq 950:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18
Error	54.29%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19
Error	54.42%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;F \leq -650000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 20
Error	58.51%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;F \leq -650000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{-98}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21
Error	62.32%
Cost	840

\[\begin{array}{l} \mathbf{if}\;F \leq -0.54:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{-98}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 22
Error	62.15%
Cost	584

\[\begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-70}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 23
Error	73.46%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -650000000:\\ \;\;\;\;-1 + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

Alternative 24
Error	70.49%
Cost	452

\[\begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

Alternative 25
Error	74.06%
Cost	388

\[\begin{array}{l} \mathbf{if}\;F \leq -5600000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

Alternative 26
Error	88.67%
Cost	192

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

Reproduce

herbie shell --seed 2023103 
(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))))))