Average Error: 13.5 → 10.8
Time: 42.4s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}\]
\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)}
↓
\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r1099830 = x;
        double r1099831 = 1.0;
        double r1099832 = B;
        double r1099833 = tan(r1099832);
        double r1099834 = r1099831 / r1099833;
        double r1099835 = r1099830 * r1099834;
        double r1099836 = -r1099835;
        double r1099837 = F;
        double r1099838 = sin(r1099832);
        double r1099839 = r1099837 / r1099838;
        double r1099840 = r1099837 * r1099837;
        double r1099841 = 2.0;
        double r1099842 = r1099840 + r1099841;
        double r1099843 = r1099841 * r1099830;
        double r1099844 = r1099842 + r1099843;
        double r1099845 = r1099831 / r1099841;
        double r1099846 = -r1099845;
        double r1099847 = pow(r1099844, r1099846);
        double r1099848 = r1099839 * r1099847;
        double r1099849 = r1099836 + r1099848;
        return r1099849;
}
↓
double f(double F, double B, double x) {
        double r1099850 = 1.0;
        double r1099851 = B;
        double r1099852 = sin(r1099851);
        double r1099853 = r1099850 / r1099852;
        double r1099854 = 2.0;
        double r1099855 = x;
        double r1099856 = F;
        double r1099857 = fma(r1099856, r1099856, r1099854);
        double r1099858 = fma(r1099854, r1099855, r1099857);
        double r1099859 = -0.5;
        double r1099860 = pow(r1099858, r1099859);
        double r1099861 = r1099860 * r1099856;
        double r1099862 = r1099853 * r1099861;
        double r1099863 = tan(r1099851);
        double r1099864 = r1099855 / r1099863;
        double r1099865 = r1099862 - r1099864;
        return r1099865;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.5

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

  2. Simplified13.4

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

  4. Applied div-inv13.5

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

  5. Applied associate-*r*10.8

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

  6. Final simplification10.8

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

Reproduce

herbie shell --seed 2019142 +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))))))