Average Error: 13.8 → 0.2
Time: 27.8s
Precision: binary64
Cost: 33096
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -10000000000:\\ \;\;\;\;\frac{\frac{\tan B}{-\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 36000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
(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 -10000000000.0)
     (/ (- (/ (tan B) (- (sin B))) x) (tan B))
     (if (<= F 36000.0)
       (- (/ (/ F (sin B)) (sqrt (fma x 2.0 (fma F F 2.0)))) 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 <= -10000000000.0) {
		tmp = ((tan(B) / -sin(B)) - x) / tan(B);
	} else if (F <= 36000.0) {
		tmp = ((F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - 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 <= -10000000000.0)
		tmp = Float64(Float64(Float64(tan(B) / Float64(-sin(B))) - x) / tan(B));
	elseif (F <= 36000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - 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, -10000000000.0], N[(N[(N[(N[Tan[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision] - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 36000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $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 -10000000000:\\
\;\;\;\;\frac{\frac{\tan B}{-\sin B} - x}{\tan B}\\

\mathbf{elif}\;F \leq 36000:\\
\;\;\;\;\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}

Error

Derivation

  1. Split input into 3 regimes

  2. if F < -1e10

    1. Initial program 25.7

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

    2. Simplified25.7

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

      [Start]25.7

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

      +-commutative [=>]25.7

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

      unsub-neg [=>]25.7

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

      +-commutative [=>]25.7

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

      *-commutative [=>]25.7

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

      fma-def [=>]25.7

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

      fma-def [=>]25.7

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

      metadata-eval [=>]25.7

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

      metadata-eval [=>]25.7

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

      associate-*r/ [=>]25.7

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

      *-rgt-identity [=>]25.7

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

    3. Taylor expanded in F around -inf 0.2

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

    4. Applied egg-rr12.3

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-\tan B\right) - \sin B \cdot \left(-x\right)}{\sin B \cdot \left(-\tan B\right)}} \]

    5. Simplified0.2

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

      [Start]12.3

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

      distribute-rgt-neg-in [<=]12.3

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

      distribute-lft-neg-in [=>]12.3

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

      div-sub [=>]12.3

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

      associate-/r* [=>]12.0

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

      mul-1-neg [=>]12.0

      \[ \frac{\frac{\color{blue}{-\left(-\tan B\right)}}{-\sin B}}{\tan B} - \frac{\sin B \cdot \left(-x\right)}{\left(-\sin B\right) \cdot \tan B} \]

      remove-double-neg [=>]12.0

      \[ \frac{\frac{\color{blue}{\tan B}}{-\sin B}}{\tan B} - \frac{\sin B \cdot \left(-x\right)}{\left(-\sin B\right) \cdot \tan B} \]

      associate-/r* [=>]0.2

      \[ \frac{\frac{\tan B}{-\sin B}}{\tan B} - \color{blue}{\frac{\frac{\sin B \cdot \left(-x\right)}{-\sin B}}{\tan B}} \]

      div-sub [<=]0.2

      \[ \color{blue}{\frac{\frac{\tan B}{-\sin B} - \frac{\sin B \cdot \left(-x\right)}{-\sin B}}{\tan B}} \]

      distribute-rgt-neg-out [=>]0.2

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

      distribute-lft-neg-out [<=]0.2

      \[ \frac{\frac{\tan B}{-\sin B} - \frac{\color{blue}{\left(-\sin B\right) \cdot x}}{-\sin B}}{\tan B} \]

      *-commutative [=>]0.2

      \[ \frac{\frac{\tan B}{-\sin B} - \frac{\color{blue}{x \cdot \left(-\sin B\right)}}{-\sin B}}{\tan B} \]

      associate-/l* [=>]0.2

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

      *-inverses [=>]0.2

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

    if -1e10 < F < 36000

    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. Simplified0.3

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

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

    3. Applied egg-rr0.3

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

    4. Applied egg-rr21.5

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

    5. Simplified0.3

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

      [Start]21.5

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

      expm1-def [=>]7.5

      \[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

      expm1-log1p [=>]0.3

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

    if 36000 < F

    1. Initial program 25.4

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

    2. Simplified25.3

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

      [Start]25.4

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

      +-commutative [=>]25.4

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

      unsub-neg [=>]25.4

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

      +-commutative [=>]25.4

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

      *-commutative [=>]25.4

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

      fma-def [=>]25.4

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

      fma-def [=>]25.4

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

      metadata-eval [=>]25.4

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

      metadata-eval [=>]25.4

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

      associate-*r/ [=>]25.3

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

      *-rgt-identity [=>]25.3

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

    3. Taylor expanded in F around inf 0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10000000000:\\ \;\;\;\;\frac{\frac{\tan B}{-\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 36000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Error0.3
Cost20744
\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\tan B}{-\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 2
Error0.7
Cost20424
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{\frac{\tan B}{-\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 1.72:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 3
Error0.7
Cost20296
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{\frac{\tan B}{-\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 4
Error0.7
Cost20040
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 5
Error0.7
Cost20040
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{\frac{\tan B}{-\sin B} - x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 6
Error7.3
Cost19908
\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -2.25 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-209}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 10500:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 7
Error7.3
Cost14480
\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.25 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 7800:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]
Alternative 8
Error7.5
Cost14288
\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.086:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]
Alternative 9
Error10.9
Cost13644
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -10500000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-37}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.0042:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 10
Error25.1
Cost13580
\[\begin{array}{l} \mathbf{if}\;F \leq -10500000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 210000000000:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 11
Error25.1
Cost13580
\[\begin{array}{l} \mathbf{if}\;F \leq -10500000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-36}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 30000000000:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 12
Error25.1
Cost13580
\[\begin{array}{l} \mathbf{if}\;F \leq -10500000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-36}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 30000000000:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 13
Error18.1
Cost13580
\[\begin{array}{l} \mathbf{if}\;F \leq -10500000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.8 \cdot 10^{-36}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 30000000000:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 14
Error29.6
Cost8080
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -10500000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-216}:\\ \;\;\;\;\frac{\frac{1}{F}}{\frac{B}{F}} - t_0\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 30000000000:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+236}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \]
Alternative 15
Error30.6
Cost7641
\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-6} \lor \neg \left(x \leq 40000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]
Alternative 16
Error35.3
Cost6856
\[\begin{array}{l} \mathbf{if}\;F \leq -1.72 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.24:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 17
Error37.9
Cost6724
\[\begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.0072:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
Alternative 18
Error42.5
Cost584
\[\begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 0.0042:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
Alternative 19
Error40.0
Cost584
\[\begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0042:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
Alternative 20
Error47.7
Cost388
\[\begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]
Alternative 21
Error56.9
Cost192
\[\frac{-1}{B} \]

Error

Reproduce

herbie shell --seed 2022354 
(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))))))