Average Error: 13.7 → 10.8
Time: 2.6m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(x, 2, \left(\mathsf{fma}\left(F, F, 2\right)\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}\]
\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(x, 2, \left(\mathsf{fma}\left(F, F, 2\right)\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r9761909 = x;
        double r9761910 = 1.0;
        double r9761911 = B;
        double r9761912 = tan(r9761911);
        double r9761913 = r9761910 / r9761912;
        double r9761914 = r9761909 * r9761913;
        double r9761915 = -r9761914;
        double r9761916 = F;
        double r9761917 = sin(r9761911);
        double r9761918 = r9761916 / r9761917;
        double r9761919 = r9761916 * r9761916;
        double r9761920 = 2.0;
        double r9761921 = r9761919 + r9761920;
        double r9761922 = r9761920 * r9761909;
        double r9761923 = r9761921 + r9761922;
        double r9761924 = r9761910 / r9761920;
        double r9761925 = -r9761924;
        double r9761926 = pow(r9761923, r9761925);
        double r9761927 = r9761918 * r9761926;
        double r9761928 = r9761915 + r9761927;
        return r9761928;
}
↓
double f(double F, double B, double x) {
        double r9761929 = 1.0;
        double r9761930 = B;
        double r9761931 = sin(r9761930);
        double r9761932 = r9761929 / r9761931;
        double r9761933 = x;
        double r9761934 = 2.0;
        double r9761935 = F;
        double r9761936 = fma(r9761935, r9761935, r9761934);
        double r9761937 = fma(r9761933, r9761934, r9761936);
        double r9761938 = -0.5;
        double r9761939 = pow(r9761937, r9761938);
        double r9761940 = r9761939 * r9761935;
        double r9761941 = r9761932 * r9761940;
        double r9761942 = tan(r9761930);
        double r9761943 = r9761933 / r9761942;
        double r9761944 = r9761941 - r9761943;
        return r9761944;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.7

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

  2. Simplified13.3

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

  4. Applied div-inv13.3

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

  5. Applied *-un-lft-identity13.3

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

  6. Applied times-frac10.8

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

  7. Simplified10.8

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

  8. Final simplification10.8

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

Reproduce

herbie shell --seed 2019120 +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))))))