Average Error: 13.2 → 0.3
Time: 37.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -4222210252.5640836:\\ \;\;\;\;\frac{\frac{F}{-\left(F + \frac{1}{F}\right)}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 3.91016330906073 \cdot 10^{+51}:\\ \;\;\;\;{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot \frac{F}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -4222210252.5640836:\\
\;\;\;\;\frac{\frac{F}{-\left(F + \frac{1}{F}\right)}}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \le 3.91016330906073 \cdot 10^{+51}:\\
\;\;\;\;{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot \frac{F}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\

\end{array}
double f(double F, double B, double x) {
        double r1033026 = x;
        double r1033027 = 1.0;
        double r1033028 = B;
        double r1033029 = tan(r1033028);
        double r1033030 = r1033027 / r1033029;
        double r1033031 = r1033026 * r1033030;
        double r1033032 = -r1033031;
        double r1033033 = F;
        double r1033034 = sin(r1033028);
        double r1033035 = r1033033 / r1033034;
        double r1033036 = r1033033 * r1033033;
        double r1033037 = 2.0;
        double r1033038 = r1033036 + r1033037;
        double r1033039 = r1033037 * r1033026;
        double r1033040 = r1033038 + r1033039;
        double r1033041 = r1033027 / r1033037;
        double r1033042 = -r1033041;
        double r1033043 = pow(r1033040, r1033042);
        double r1033044 = r1033035 * r1033043;
        double r1033045 = r1033032 + r1033044;
        return r1033045;
}
↓
double f(double F, double B, double x) {
        double r1033046 = F;
        double r1033047 = -4222210252.5640836;
        bool r1033048 = r1033046 <= r1033047;
        double r1033049 = 1.0;
        double r1033050 = r1033049 / r1033046;
        double r1033051 = r1033046 + r1033050;
        double r1033052 = -r1033051;
        double r1033053 = r1033046 / r1033052;
        double r1033054 = B;
        double r1033055 = sin(r1033054);
        double r1033056 = r1033053 / r1033055;
        double r1033057 = x;
        double r1033058 = tan(r1033054);
        double r1033059 = r1033057 / r1033058;
        double r1033060 = r1033056 - r1033059;
        double r1033061 = 3.91016330906073e+51;
        bool r1033062 = r1033046 <= r1033061;
        double r1033063 = 2.0;
        double r1033064 = r1033057 * r1033063;
        double r1033065 = r1033046 * r1033046;
        double r1033066 = r1033063 + r1033065;
        double r1033067 = r1033064 + r1033066;
        double r1033068 = -r1033049;
        double r1033069 = r1033068 / r1033063;
        double r1033070 = pow(r1033067, r1033069);
        double r1033071 = r1033046 / r1033055;
        double r1033072 = r1033070 * r1033071;
        double r1033073 = cos(r1033054);
        double r1033074 = r1033073 * r1033057;
        double r1033075 = r1033074 / r1033055;
        double r1033076 = r1033072 - r1033075;
        double r1033077 = r1033046 / r1033051;
        double r1033078 = r1033077 / r1033055;
        double r1033079 = r1033078 - r1033059;
        double r1033080 = r1033062 ? r1033076 : r1033079;
        double r1033081 = r1033048 ? r1033060 : r1033080;
        return r1033081;
}

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 < -4222210252.5640836

    1. Initial program 24.7

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

    2. Simplified24.7

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

    4. Applied pow-neg24.7

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

    5. Applied frac-times19.3

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

    6. Simplified19.3

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

    8. Applied associate-/r*19.3

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

    9. Taylor expanded around -inf 0.1

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

    if -4222210252.5640836 < F < 3.91016330906073e+51

    1. Initial program 0.5

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

    2. Simplified0.4

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

    3. Taylor expanded around inf 0.4

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

    if 3.91016330906073e+51 < F

    1. Initial program 28.2

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

    2. Simplified28.1

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

    4. Applied pow-neg28.1

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

    5. Applied frac-times22.1

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

    6. Simplified22.1

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

    8. Applied associate-/r*22.1

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

    9. Taylor expanded around inf 0.1

      \[\leadsto \frac{\frac{F}{\color{blue}{F + \frac{1}{F}}}}{\sin B} - \frac{x}{\tan B}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -4222210252.5640836:\\ \;\;\;\;\frac{\frac{F}{-\left(F + \frac{1}{F}\right)}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 3.91016330906073 \cdot 10^{+51}:\\ \;\;\;\;{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot \frac{F}{\sin B} - \frac{\cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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))))))