Average Error: 14.3 → 0.3
Time: 10.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1683736694.52556157:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\ \mathbf{elif}\;F \le 8.6863795197068081 \cdot 10^{53}:\\ \;\;\;\;\mathsf{fma}\left(F \cdot \frac{1}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B} + \frac{x}{\sin B \cdot {F}^{2}}, \frac{1}{\sin B}\right)\\ \end{array}\]
\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 -1683736694.52556157:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\

\mathbf{elif}\;F \le 8.6863795197068081 \cdot 10^{53}:\\
\;\;\;\;\mathsf{fma}\left(F \cdot \frac{1}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\\

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

\end{array}
double f(double F, double B, double x) {
        double r31605 = x;
        double r31606 = 1.0;
        double r31607 = B;
        double r31608 = tan(r31607);
        double r31609 = r31606 / r31608;
        double r31610 = r31605 * r31609;
        double r31611 = -r31610;
        double r31612 = F;
        double r31613 = sin(r31607);
        double r31614 = r31612 / r31613;
        double r31615 = r31612 * r31612;
        double r31616 = 2.0;
        double r31617 = r31615 + r31616;
        double r31618 = r31616 * r31605;
        double r31619 = r31617 + r31618;
        double r31620 = r31606 / r31616;
        double r31621 = -r31620;
        double r31622 = pow(r31619, r31621);
        double r31623 = r31614 * r31622;
        double r31624 = r31611 + r31623;
        return r31624;
}
↓
double f(double F, double B, double x) {
        double r31625 = F;
        double r31626 = -1683736694.5255616;
        bool r31627 = r31625 <= r31626;
        double r31628 = 1.0;
        double r31629 = x;
        double r31630 = B;
        double r31631 = sin(r31630);
        double r31632 = 2.0;
        double r31633 = pow(r31625, r31632);
        double r31634 = r31631 * r31633;
        double r31635 = r31629 / r31634;
        double r31636 = cos(r31630);
        double r31637 = r31629 * r31636;
        double r31638 = r31637 / r31631;
        double r31639 = 1.0;
        double r31640 = r31639 / r31631;
        double r31641 = fma(r31628, r31638, r31640);
        double r31642 = -r31641;
        double r31643 = fma(r31628, r31635, r31642);
        double r31644 = 8.686379519706808e+53;
        bool r31645 = r31625 <= r31644;
        double r31646 = r31625 * r31640;
        double r31647 = r31625 * r31625;
        double r31648 = 2.0;
        double r31649 = r31647 + r31648;
        double r31650 = r31648 * r31629;
        double r31651 = r31649 + r31650;
        double r31652 = r31628 / r31648;
        double r31653 = -r31652;
        double r31654 = pow(r31651, r31653);
        double r31655 = r31628 * r31638;
        double r31656 = -r31655;
        double r31657 = fma(r31646, r31654, r31656);
        double r31658 = -r31628;
        double r31659 = r31638 + r31635;
        double r31660 = fma(r31658, r31659, r31640);
        double r31661 = r31645 ? r31657 : r31660;
        double r31662 = r31627 ? r31643 : r31661;
        return r31662;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -1683736694.5255616

    1. Initial program 25.8

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

    2. Simplified25.8

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -1683736694.5255616 < F < 8.686379519706808e+53

    1. Initial program 0.5

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

    2. Simplified0.5

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

    4. Applied associate-*r/0.4

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

    6. Applied div-inv0.5

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

    7. Taylor expanded around inf 0.5

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

    if 8.686379519706808e+53 < F

    1. Initial program 29.8

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

    2. Simplified29.8

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1683736694.52556157:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\ \mathbf{elif}\;F \le 8.6863795197068081 \cdot 10^{53}:\\ \;\;\;\;\mathsf{fma}\left(F \cdot \frac{1}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -1 \cdot \frac{x \cdot \cos B}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B} + \frac{x}{\sin B \cdot {F}^{2}}, \frac{1}{\sin B}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))))