Average Error: 14.1 → 0.3
Time: 13.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1.40065760476294 \cdot 10^{108}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{F}{{\left(\frac{-1}{F}\right)}^{-1} + 1 \cdot {\left(\frac{1}{{-1}^{1} \cdot {F}^{1}}\right)}^{1}}}{\sin B}\\ \mathbf{elif}\;F \le 760094393.57388496:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{\frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -1.40065760476294 \cdot 10^{108}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{F}{{\left(\frac{-1}{F}\right)}^{-1} + 1 \cdot {\left(\frac{1}{{-1}^{1} \cdot {F}^{1}}\right)}^{1}}}{\sin B}\\

\mathbf{elif}\;F \le 760094393.57388496:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{\frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}{\sin B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r44895 = x;
        double r44896 = 1.0;
        double r44897 = B;
        double r44898 = tan(r44897);
        double r44899 = r44896 / r44898;
        double r44900 = r44895 * r44899;
        double r44901 = -r44900;
        double r44902 = F;
        double r44903 = sin(r44897);
        double r44904 = r44902 / r44903;
        double r44905 = r44902 * r44902;
        double r44906 = 2.0;
        double r44907 = r44905 + r44906;
        double r44908 = r44906 * r44895;
        double r44909 = r44907 + r44908;
        double r44910 = r44896 / r44906;
        double r44911 = -r44910;
        double r44912 = pow(r44909, r44911);
        double r44913 = r44904 * r44912;
        double r44914 = r44901 + r44913;
        return r44914;
}
↓
double f(double F, double B, double x) {
        double r44915 = F;
        double r44916 = -1.40065760476294e+108;
        bool r44917 = r44915 <= r44916;
        double r44918 = x;
        double r44919 = 1.0;
        double r44920 = r44918 * r44919;
        double r44921 = B;
        double r44922 = tan(r44921);
        double r44923 = r44920 / r44922;
        double r44924 = -r44923;
        double r44925 = -1.0;
        double r44926 = r44925 / r44915;
        double r44927 = -1.0;
        double r44928 = pow(r44926, r44927);
        double r44929 = 1.0;
        double r44930 = pow(r44925, r44919);
        double r44931 = pow(r44915, r44919);
        double r44932 = r44930 * r44931;
        double r44933 = r44929 / r44932;
        double r44934 = pow(r44933, r44919);
        double r44935 = r44919 * r44934;
        double r44936 = r44928 + r44935;
        double r44937 = r44915 / r44936;
        double r44938 = sin(r44921);
        double r44939 = r44937 / r44938;
        double r44940 = r44924 + r44939;
        double r44941 = 760094393.573885;
        bool r44942 = r44915 <= r44941;
        double r44943 = cos(r44921);
        double r44944 = r44918 * r44943;
        double r44945 = r44944 / r44938;
        double r44946 = r44919 * r44945;
        double r44947 = -r44946;
        double r44948 = r44915 * r44915;
        double r44949 = 2.0;
        double r44950 = r44948 + r44949;
        double r44951 = r44949 * r44918;
        double r44952 = r44950 + r44951;
        double r44953 = r44919 / r44949;
        double r44954 = pow(r44952, r44953);
        double r44955 = r44915 / r44954;
        double r44956 = r44955 / r44938;
        double r44957 = r44947 + r44956;
        double r44958 = r44929 / r44938;
        double r44959 = 2.0;
        double r44960 = pow(r44915, r44959);
        double r44961 = r44938 * r44960;
        double r44962 = r44929 / r44961;
        double r44963 = r44919 * r44962;
        double r44964 = r44958 - r44963;
        double r44965 = r44924 + r44964;
        double r44966 = r44942 ? r44957 : r44965;
        double r44967 = r44917 ? r44940 : r44966;
        return r44967;
}

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 < -1.40065760476294e+108

    1. Initial program 33.8

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

    3. Applied associate-*l/27.6

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

    5. Applied associate-*r/27.5

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

    7. Applied pow-neg27.5

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

    8. Applied un-div-inv27.5

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

    9. Taylor expanded around -inf 0.2

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

    if -1.40065760476294e+108 < F < 760094393.573885

    1. Initial program 1.0

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

    3. Applied associate-*l/0.4

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

    5. Applied associate-*r/0.3

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

    7. Applied pow-neg0.3

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

    8. Applied un-div-inv0.3

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

    9. Taylor expanded around inf 0.3

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

    if 760094393.573885 < F

    1. Initial program 26.8

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

    3. Applied associate-*l/20.8

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

    5. Applied associate-*r/20.8

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

    7. Applied pow-neg20.7

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

    8. Applied un-div-inv20.7

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

    9. Taylor expanded around inf 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.40065760476294 \cdot 10^{108}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{F}{{\left(\frac{-1}{F}\right)}^{-1} + 1 \cdot {\left(\frac{1}{{-1}^{1} \cdot {F}^{1}}\right)}^{1}}}{\sin B}\\ \mathbf{elif}\;F \le 760094393.57388496:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{\frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(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))))))