Average Error: 13.8 → 0.3
Time: 32.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -4.183187481012982 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 6.013802817876398 \cdot 10^{+19}:\\ \;\;\;\;\frac{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\frac{-1}{2}}}{\frac{1}{\frac{F}{\sin B}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot \left(F \cdot F\right)}\right) - \frac{x}{\tan B}\\ \end{array}\]
\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 -4.183187481012982 \cdot 10^{+22}:\\
\;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\

\mathbf{elif}\;F \le 6.013802817876398 \cdot 10^{+19}:\\
\;\;\;\;\frac{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\frac{-1}{2}}}{\frac{1}{\frac{F}{\sin B}}} - \frac{x}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r817803 = x;
        double r817804 = 1.0;
        double r817805 = B;
        double r817806 = tan(r817805);
        double r817807 = r817804 / r817806;
        double r817808 = r817803 * r817807;
        double r817809 = -r817808;
        double r817810 = F;
        double r817811 = sin(r817805);
        double r817812 = r817810 / r817811;
        double r817813 = r817810 * r817810;
        double r817814 = 2.0;
        double r817815 = r817813 + r817814;
        double r817816 = r817814 * r817803;
        double r817817 = r817815 + r817816;
        double r817818 = r817804 / r817814;
        double r817819 = -r817818;
        double r817820 = pow(r817817, r817819);
        double r817821 = r817812 * r817820;
        double r817822 = r817809 + r817821;
        return r817822;
}
↓
double f(double F, double B, double x) {
        double r817823 = F;
        double r817824 = -4.183187481012982e+22;
        bool r817825 = r817823 <= r817824;
        double r817826 = 1.0;
        double r817827 = B;
        double r817828 = sin(r817827);
        double r817829 = r817823 * r817823;
        double r817830 = r817828 * r817829;
        double r817831 = r817826 / r817830;
        double r817832 = r817826 / r817828;
        double r817833 = r817831 - r817832;
        double r817834 = x;
        double r817835 = tan(r817827);
        double r817836 = r817834 / r817835;
        double r817837 = r817833 - r817836;
        double r817838 = 6.013802817876398e+19;
        bool r817839 = r817823 <= r817838;
        double r817840 = 2.0;
        double r817841 = r817834 * r817840;
        double r817842 = r817840 + r817829;
        double r817843 = r817841 + r817842;
        double r817844 = -0.5;
        double r817845 = pow(r817843, r817844);
        double r817846 = r817823 / r817828;
        double r817847 = r817826 / r817846;
        double r817848 = r817845 / r817847;
        double r817849 = r817848 - r817836;
        double r817850 = r817832 - r817831;
        double r817851 = r817850 - r817836;
        double r817852 = r817839 ? r817849 : r817851;
        double r817853 = r817825 ? r817837 : r817852;
        return r817853;
}

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 < -4.183187481012982e+22

    1. Initial program 26.4

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

    2. Simplified25.5

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

    if -4.183187481012982e+22 < F < 6.013802817876398e+19

    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

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

    4. Applied clear-num0.4

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

    if 6.013802817876398e+19 < F

    1. Initial program 26.6

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

    2. Simplified25.8

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -4.183187481012982 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 6.013802817876398 \cdot 10^{+19}:\\ \;\;\;\;\frac{{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\frac{-1}{2}}}{\frac{1}{\frac{F}{\sin B}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot \left(F \cdot F\right)}\right) - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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))))))