Average Error: 13.5 → 0.2
Time: 1.6m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -45183016.350486509501934051513671875:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 4.890764603747722816251553012989461421967:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]
\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 -45183016.350486509501934051513671875:\\
\;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 4.890764603747722816251553012989461421967:\\
\;\;\;\;\frac{\frac{F}{\sin B}}{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r4275964 = x;
        double r4275965 = 1.0;
        double r4275966 = B;
        double r4275967 = tan(r4275966);
        double r4275968 = r4275965 / r4275967;
        double r4275969 = r4275964 * r4275968;
        double r4275970 = -r4275969;
        double r4275971 = F;
        double r4275972 = sin(r4275966);
        double r4275973 = r4275971 / r4275972;
        double r4275974 = r4275971 * r4275971;
        double r4275975 = 2.0;
        double r4275976 = r4275974 + r4275975;
        double r4275977 = r4275975 * r4275964;
        double r4275978 = r4275976 + r4275977;
        double r4275979 = r4275965 / r4275975;
        double r4275980 = -r4275979;
        double r4275981 = pow(r4275978, r4275980);
        double r4275982 = r4275973 * r4275981;
        double r4275983 = r4275970 + r4275982;
        return r4275983;
}
↓
double f(double F, double B, double x) {
        double r4275984 = F;
        double r4275985 = -45183016.35048651;
        bool r4275986 = r4275984 <= r4275985;
        double r4275987 = 1.0;
        double r4275988 = B;
        double r4275989 = sin(r4275988);
        double r4275990 = r4275987 / r4275989;
        double r4275991 = r4275984 * r4275984;
        double r4275992 = r4275990 / r4275991;
        double r4275993 = 1.0;
        double r4275994 = r4275993 / r4275989;
        double r4275995 = r4275992 - r4275994;
        double r4275996 = x;
        double r4275997 = r4275996 * r4275987;
        double r4275998 = tan(r4275988);
        double r4275999 = r4275997 / r4275998;
        double r4276000 = r4275995 - r4275999;
        double r4276001 = 4.890764603747723;
        bool r4276002 = r4275984 <= r4276001;
        double r4276003 = r4275984 / r4275989;
        double r4276004 = 2.0;
        double r4276005 = r4276004 * r4275996;
        double r4276006 = r4276004 + r4275991;
        double r4276007 = r4276005 + r4276006;
        double r4276008 = r4275987 / r4276004;
        double r4276009 = pow(r4276007, r4276008);
        double r4276010 = r4276003 / r4276009;
        double r4276011 = r4276010 - r4275999;
        double r4276012 = r4275994 - r4275992;
        double r4276013 = r4276012 - r4275999;
        double r4276014 = r4276002 ? r4276011 : r4276013;
        double r4276015 = r4275986 ? r4276000 : r4276014;
        return r4276015;
}

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 < -45183016.35048651

    1. Initial program 25.6

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

    2. Simplified20.0

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

    if -45183016.35048651 < F < 4.890764603747723

    1. Initial program 0.4

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

    2. Simplified0.3

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

    4. Applied pow-neg0.3

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

    5. Applied associate-/r/0.3

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

    6. Applied associate-/r*0.3

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

    7. Simplified0.3

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

    if 4.890764603747723 < F

    1. Initial program 23.4

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

    2. Simplified17.6

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -45183016.350486509501934051513671875:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 4.890764603747722816251553012989461421967:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))