Average Error: 13.9 → 0.4
Time: 33.5s
Precision: binary64
Cost: 33160
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\ \end{array} \]
(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 -2.5e+59)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2e+73)
       (- (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ (sin B) F)) t_0)
       (- (/ 1.0 (sin B)) (pow (/ (tan B) x) -1.0))))))
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 <= -2.5e+59) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2e+73) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / (sin(B) / F)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - pow((tan(B) / x), -1.0);
	}
	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 <= -2.5e+59)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2e+73)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / Float64(sin(B) / F)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - (Float64(tan(B) / x) ^ -1.0));
	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, -2.5e+59], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2e+73], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[Power[N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision]), $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 -2.5 \cdot 10^{+59}:\\
\;\;\;\;\frac{-1}{\sin B} - t_0\\

\mathbf{elif}\;F \leq 2 \cdot 10^{+73}:\\
\;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - t_0\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if F < -2.4999999999999999e59

    1. Initial program 29.9

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

    2. Simplified29.8

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

      [Start]29.9

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

      +-commutative [=>]29.9

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

      unsub-neg [=>]29.9

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

      +-commutative [=>]29.9

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

      *-commutative [=>]29.9

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

      fma-def [=>]29.9

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

      fma-def [=>]29.9

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

      metadata-eval [=>]29.9

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

      metadata-eval [=>]29.9

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

      associate-*r/ [=>]29.8

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

      *-rgt-identity [=>]29.8

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

    3. Taylor expanded in F around -inf 0.2

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

    if -2.4999999999999999e59 < F < 1.99999999999999997e73

    1. Initial program 0.9

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

    2. Simplified0.8

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

      [Start]0.9

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

      +-commutative [=>]0.9

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

      unsub-neg [=>]0.9

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

      +-commutative [=>]0.9

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

      *-commutative [=>]0.9

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

      fma-def [=>]0.9

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

      fma-def [=>]0.9

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

      metadata-eval [=>]0.9

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

      metadata-eval [=>]0.9

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

      associate-*r/ [=>]0.8

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

      *-rgt-identity [=>]0.8

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

    3. Applied egg-rr0.6

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

    if 1.99999999999999997e73 < F

    1. Initial program 30.2

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

    2. Simplified30.1

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

      [Start]30.2

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

      +-commutative [=>]30.2

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

      unsub-neg [=>]30.2

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

      +-commutative [=>]30.2

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

      *-commutative [=>]30.2

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

      fma-def [=>]30.2

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

      fma-def [=>]30.2

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

      metadata-eval [=>]30.2

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

      metadata-eval [=>]30.2

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

      associate-*r/ [=>]30.1

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

      *-rgt-identity [=>]30.1

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

    3. Taylor expanded in F around inf 0.1

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

    4. Applied egg-rr0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error0.4
Cost33160
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -100000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\ \end{array} \]
Alternative 2
Error0.6
Cost27272
\[\begin{array}{l} \mathbf{if}\;F \leq -45000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{+88}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{\frac{x}{\sin B}}{\frac{1}{\cos B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\ \end{array} \]
Alternative 3
Error0.5
Cost27144
\[\begin{array}{l} \mathbf{if}\;F \leq -100000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\ \end{array} \]
Alternative 4
Error0.3
Cost20744
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -45000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 130:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 5
Error0.7
Cost20424
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2600:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.92:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 6
Error0.7
Cost20168
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 7
Error4.9
Cost14480
\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.045:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 20:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]
Alternative 8
Error8.1
Cost14288
\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0022:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.12:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \]
Alternative 9
Error5.1
Cost14288
\[\begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00048:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + t_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 0.6:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_2\\ \end{array} \]
Alternative 10
Error10.3
Cost13644
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \]
Alternative 11
Error15.3
Cost13380
\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.000105:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{1}{B} - t_0}}\\ \end{array} \]
Alternative 12
Error24.7
Cost13316
\[\begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+101}:\\ \;\;\;\;-\frac{\cos B}{\frac{\sin B}{x}}\\ \mathbf{elif}\;F \leq -1080000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{1}{B} - \frac{x}{\tan B}}}\\ \end{array} \]
Alternative 13
Error24.7
Cost13316
\[\begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{\frac{-\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq -1080000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{1}{B} - \frac{x}{\tan B}}}\\ \end{array} \]
Alternative 14
Error24.7
Cost13316
\[\begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq -1080000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{1}{B} - \frac{x}{\tan B}}}\\ \end{array} \]
Alternative 15
Error24.7
Cost7500
\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1100000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{1}{B} - \frac{x}{\tan B}}}\\ \end{array} \]
Alternative 16
Error24.8
Cost7244
\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1450000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
Alternative 17
Error26.7
Cost7052
\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1800000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10800000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 18
Error33.4
Cost6920
\[\begin{array}{l} \mathbf{if}\;F \leq -1080000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-28}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 19
Error34.2
Cost6856
\[\begin{array}{l} \mathbf{if}\;F \leq -1080000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
Alternative 20
Error36.7
Cost6724
\[\begin{array}{l} \mathbf{if}\;F \leq -1080000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10800000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]
Alternative 21
Error39.7
Cost968
\[\begin{array}{l} \mathbf{if}\;F \leq -0.046:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 10800000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]
Alternative 22
Error39.6
Cost840
\[\begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
Alternative 23
Error39.6
Cost584
\[\begin{array}{l} \mathbf{if}\;F \leq -0.046:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
Alternative 24
Error44.4
Cost520
\[\begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 185000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
Alternative 25
Error42.0
Cost520
\[\begin{array}{l} \mathbf{if}\;F \leq -0.046:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 185000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
Alternative 26
Error51.7
Cost324
\[\begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
Alternative 27
Error56.7
Cost192
\[\frac{-1}{B} \]

Error

Reproduce

herbie shell --seed 2023012 
(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))))))