Average Error: 13.5 → 0.2
Time: 41.6s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -51018.23700738415936939418315887451171875:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{-1}{\sin B} + \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \mathbf{elif}\;F \le 13415.64195421903423266485333442687988281:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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}
\mathbf{if}\;F \le -51018.23700738415936939418315887451171875:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{-1}{\sin B} + \frac{1}{\sin B \cdot {F}^{2}}\right)\\

\mathbf{elif}\;F \le 13415.64195421903423266485333442687988281:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\

\end{array}
double f(double F, double B, double x) {
        double r103588 = x;
        double r103589 = 1.0;
        double r103590 = B;
        double r103591 = tan(r103590);
        double r103592 = r103589 / r103591;
        double r103593 = r103588 * r103592;
        double r103594 = -r103593;
        double r103595 = F;
        double r103596 = sin(r103590);
        double r103597 = r103595 / r103596;
        double r103598 = r103595 * r103595;
        double r103599 = 2.0;
        double r103600 = r103598 + r103599;
        double r103601 = r103599 * r103588;
        double r103602 = r103600 + r103601;
        double r103603 = r103589 / r103599;
        double r103604 = -r103603;
        double r103605 = pow(r103602, r103604);
        double r103606 = r103597 * r103605;
        double r103607 = r103594 + r103606;
        return r103607;
}
↓
double f(double F, double B, double x) {
        double r103608 = F;
        double r103609 = -51018.23700738416;
        bool r103610 = r103608 <= r103609;
        double r103611 = x;
        double r103612 = 1.0;
        double r103613 = r103611 * r103612;
        double r103614 = B;
        double r103615 = tan(r103614);
        double r103616 = r103613 / r103615;
        double r103617 = -r103616;
        double r103618 = -1.0;
        double r103619 = sin(r103614);
        double r103620 = r103618 / r103619;
        double r103621 = 2.0;
        double r103622 = pow(r103608, r103621);
        double r103623 = r103619 * r103622;
        double r103624 = r103612 / r103623;
        double r103625 = r103620 + r103624;
        double r103626 = r103617 + r103625;
        double r103627 = 13415.641954219034;
        bool r103628 = r103608 <= r103627;
        double r103629 = cos(r103614);
        double r103630 = r103611 * r103629;
        double r103631 = r103630 / r103619;
        double r103632 = r103612 * r103631;
        double r103633 = -r103632;
        double r103634 = r103608 / r103619;
        double r103635 = r103608 * r103608;
        double r103636 = 2.0;
        double r103637 = r103635 + r103636;
        double r103638 = r103636 * r103611;
        double r103639 = r103637 + r103638;
        double r103640 = r103612 / r103636;
        double r103641 = -r103640;
        double r103642 = pow(r103639, r103641);
        double r103643 = r103634 * r103642;
        double r103644 = r103633 + r103643;
        double r103645 = 1.0;
        double r103646 = r103645 / r103619;
        double r103647 = r103646 - r103624;
        double r103648 = r103617 + r103647;
        double r103649 = r103628 ? r103644 : r103648;
        double r103650 = r103610 ? r103626 : r103649;
        return r103650;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if F < -51018.23700738416

    1. Initial program 24.1

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

    3. Applied pow-neg24.1

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

    4. Applied frac-times18.5

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

    5. Simplified18.5

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

    6. Simplified18.5

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

    8. Applied associate-*r/18.5

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

    9. Taylor expanded around -inf 0.2

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

    10. Simplified0.2

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

    if -51018.23700738416 < F < 13415.641954219034

    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. Taylor expanded around inf 0.3

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

    if 13415.641954219034 < F

    1. Initial program 25.3

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

    3. Applied pow-neg25.3

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

    4. Applied frac-times19.8

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

    5. Simplified19.8

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

    6. Simplified19.8

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

    8. Applied associate-*r/19.7

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

    9. Taylor expanded around inf 0.2

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

    10. Simplified0.2

      \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -51018.23700738415936939418315887451171875:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{-1}{\sin B} + \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \mathbf{elif}\;F \le 13415.64195421903423266485333442687988281:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))