Average Error: 13.3 → 11.0
Time: 43.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\left(\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot F\right) \cdot \frac{1}{\sqrt[3]{\sin B}} - \frac{x}{\tan B}\]
\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)}
↓
\left(\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot F\right) \cdot \frac{1}{\sqrt[3]{\sin B}} - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r1408833 = x;
        double r1408834 = 1.0;
        double r1408835 = B;
        double r1408836 = tan(r1408835);
        double r1408837 = r1408834 / r1408836;
        double r1408838 = r1408833 * r1408837;
        double r1408839 = -r1408838;
        double r1408840 = F;
        double r1408841 = sin(r1408835);
        double r1408842 = r1408840 / r1408841;
        double r1408843 = r1408840 * r1408840;
        double r1408844 = 2.0;
        double r1408845 = r1408843 + r1408844;
        double r1408846 = r1408844 * r1408833;
        double r1408847 = r1408845 + r1408846;
        double r1408848 = r1408834 / r1408844;
        double r1408849 = -r1408848;
        double r1408850 = pow(r1408847, r1408849);
        double r1408851 = r1408842 * r1408850;
        double r1408852 = r1408839 + r1408851;
        return r1408852;
}
↓
double f(double F, double B, double x) {
        double r1408853 = x;
        double r1408854 = 2.0;
        double r1408855 = F;
        double r1408856 = fma(r1408855, r1408855, r1408854);
        double r1408857 = fma(r1408853, r1408854, r1408856);
        double r1408858 = -0.5;
        double r1408859 = pow(r1408857, r1408858);
        double r1408860 = B;
        double r1408861 = sin(r1408860);
        double r1408862 = cbrt(r1408861);
        double r1408863 = r1408862 * r1408862;
        double r1408864 = r1408859 / r1408863;
        double r1408865 = r1408864 * r1408855;
        double r1408866 = 1.0;
        double r1408867 = r1408866 / r1408862;
        double r1408868 = r1408865 * r1408867;
        double r1408869 = tan(r1408860);
        double r1408870 = r1408853 / r1408869;
        double r1408871 = r1408868 - r1408870;
        return r1408871;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.3

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

  2. Simplified13.2

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

  4. Applied div-inv13.2

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

  5. Applied associate-*r*10.7

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

  7. Applied add-cube-cbrt11.0

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

  8. Applied *-un-lft-identity11.0

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

  9. Applied times-frac11.0

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

  10. Applied associate-*r*11.0

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

  11. Simplified11.0

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

  12. Final simplification11.0

    \[\leadsto \left(\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot F\right) \cdot \frac{1}{\sqrt[3]{\sin B}} - \frac{x}{\tan B}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))