?

Average Accuracy: 79.0% → 99.6%
Time: 38.7s
Precision: binary64
Cost: 33160

?

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
(FPCore (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+32)
     (- (/ -1.0 (sin B)) (/ (* (cos B) x) (sin B)))
     (if (<= F 5.5e+81)
       (- (/ F (/ (sin B) (pow (fma x 2.0 (fma F F 2.0)) -0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))
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+32) {
		tmp = (-1.0 / sin(B)) - ((cos(B) * x) / sin(B));
	} else if (F <= 5.5e+81) {
		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;
}

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+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(cos(B) * x) / sin(B)));
	elseif (F <= 5.5e+81)
		tmp = Float64(Float64(F / Float64(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

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+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e+81], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

\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^{+32}:\\
\;\;\;\;\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\

\mathbf{elif}\;F \leq 5.5 \cdot 10^{+81}:\\
\;\;\;\;\frac{F}{\frac{\sin B}{{\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}

Error?

Derivation?

  1. Split input into 3 regimes

  2. if F < -1.00000000000000005e32

    1. Initial program 58.0%

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

    2. Simplified58.1%

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

      [Start]58.0

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

      +-commutative [=>]58.0

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

      unsub-neg [=>]58.0

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

      +-commutative [=>]58.0

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

      *-commutative [=>]58.0

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

      fma-def [=>]58.0

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

      fma-def [=>]58.0

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

      metadata-eval [=>]58.0

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

      metadata-eval [=>]58.0

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

      associate-*r/ [=>]58.1

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

      *-rgt-identity [=>]58.1

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

    3. Taylor expanded in F around -inf 99.8%

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

    4. Taylor expanded in x around 0 99.7%

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

    if -1.00000000000000005e32 < F < 5.5000000000000003e81

    1. Initial program 98.5%

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

    2. Simplified98.6%

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

      [Start]98.5

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

      +-commutative [=>]98.5

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

      unsub-neg [=>]98.5

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

      +-commutative [=>]98.5

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

      *-commutative [=>]98.5

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

      fma-def [=>]98.5

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

      fma-def [=>]98.5

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

      metadata-eval [=>]98.5

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

      metadata-eval [=>]98.5

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

      associate-*r/ [=>]98.6

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

      *-rgt-identity [=>]98.6

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

    3. Applied egg-rr99.5%

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

      [Start]98.6

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

      associate-*l/ [=>]99.6

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

      associate-/l* [=>]99.5

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

    if 5.5000000000000003e81 < F

    1. Initial program 51.4%

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

    2. Simplified51.5%

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

      [Start]51.4

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

      +-commutative [=>]51.4

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

      unsub-neg [=>]51.4

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

      +-commutative [=>]51.4

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

      *-commutative [=>]51.4

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

      fma-def [=>]51.4

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

      fma-def [=>]51.4

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

      metadata-eval [=>]51.4

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

      metadata-eval [=>]51.4

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

      associate-*r/ [=>]51.5

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

      *-rgt-identity [=>]51.5

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

    3. Taylor expanded in F around inf 99.8%

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

  4. Final simplification99.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy99.4%
Cost20616
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 27:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 2
Accuracy99.1%
Cost20424
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.82:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 3
Accuracy99.1%
Cost20040
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 4
Accuracy99.1%
Cost20040
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 5
Accuracy99.1%
Cost20040
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 6
Accuracy89.8%
Cost14472
\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -1.15 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.42 \cdot 10^{-204}:\\ \;\;\;\;x \cdot \frac{-1}{B} + t_0 \cdot \left(F \cdot t_2\right)\\ \mathbf{elif}\;F \leq 18:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \]
Alternative 7
Accuracy89.4%
Cost14348
\[\begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-207}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 22:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]
Alternative 8
Accuracy89.4%
Cost14216
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.22:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 9
Accuracy87.4%
Cost14160
\[\begin{array}{l} t_0 := \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-162}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]
Alternative 10
Accuracy89.9%
Cost14028
\[\begin{array}{l} t_0 := \sqrt{2 + x \cdot 2}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{-204}:\\ \;\;\;\;\frac{F}{\sin B \cdot t_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.33:\\ \;\;\;\;\frac{F}{B \cdot t_0} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]
Alternative 11
Accuracy62.5%
Cost13713
\[\begin{array}{l} \mathbf{if}\;F \leq -0.04:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 40 \lor \neg \left(F \leq 5.5 \cdot 10^{+137}\right) \land F \leq 1.1 \cdot 10^{+201}:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \end{array} \]
Alternative 12
Accuracy62.5%
Cost13713
\[\begin{array}{l} \mathbf{if}\;F \leq -0.04:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 116:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \end{array} \]
Alternative 13
Accuracy72.8%
Cost13713
\[\begin{array}{l} \mathbf{if}\;F \leq -1.72 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 125:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \end{array} \]
Alternative 14
Accuracy81.4%
Cost13512
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-105}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 15
Accuracy56.2%
Cost8592
\[\begin{array}{l} t_0 := x \cdot \frac{-1}{B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right)\\ \mathbf{if}\;F \leq -65:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{-1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 27:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 16
Accuracy55.9%
Cost8400
\[\begin{array}{l} t_0 := x \cdot \frac{-1}{B} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(F \cdot \left(\frac{1}{B} + B \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;F \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{-1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 0.105:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 17
Accuracy55.9%
Cost8400
\[\begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := x \cdot \frac{-1}{B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-288}:\\ \;\;\;\;t_1 + t_0 \cdot \left(F \cdot \left(\frac{1}{B} + B \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;\frac{-1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 0.084:\\ \;\;\;\;t_1 + t_0 \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right)\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.15 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 18
Accuracy55.9%
Cost8208
\[\begin{array}{l} t_0 := x \cdot \frac{-1}{B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-288}:\\ \;\;\;\;t_0 + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(F \cdot \left(\frac{1}{B} + B \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{-1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 8.2:\\ \;\;\;\;t_0 + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 19
Accuracy51.5%
Cost7769
\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.018:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-288}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 27.5:\\ \;\;\;\;\frac{-1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 20
Accuracy51.4%
Cost7769
\[\begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.04:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{\frac{1}{t_0}}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-288}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 27.5:\\ \;\;\;\;\frac{-1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 21
Accuracy55.9%
Cost7769
\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1}{B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 0.175:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+137} \lor \neg \left(F \leq 1.1 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 22
Accuracy50.5%
Cost7113
\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-46} \lor \neg \left(x \leq 8.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
Alternative 23
Accuracy41.4%
Cost6724
\[\begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{B} + \left(\frac{1}{B} + B \cdot 0.16666666666666666\right)\\ \end{array} \]
Alternative 24
Accuracy37.3%
Cost1096
\[\begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{B} + \left(\frac{1}{B} + B \cdot 0.16666666666666666\right)\\ \end{array} \]
Alternative 25
Accuracy29.4%
Cost836
\[\begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \end{array} \]
Alternative 26
Accuracy29.3%
Cost708
\[\begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \end{array} \]
Alternative 27
Accuracy29.6%
Cost452
\[\begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
Alternative 28
Accuracy26.1%
Cost388
\[\begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
Alternative 29
Accuracy10.9%
Cost192
\[\frac{-1}{B} \]

Error

Reproduce?

herbie shell --seed 2023137 
(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))))))