Average Error: 13.6 → 0.3
Time: 32.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -2.034324775890907 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\left(-\frac{\sin B}{F}\right) \cdot \frac{x}{F} - \sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 107303683.05601409:\\ \;\;\;\;\frac{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B + \frac{x}{F \cdot F} \cdot \sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -2.034324775890907 \cdot 10^{+59}:\\
\;\;\;\;\frac{1}{\left(-\frac{\sin B}{F}\right) \cdot \frac{x}{F} - \sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \le 107303683.05601409:\\
\;\;\;\;\frac{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r1497627 = x;
        double r1497628 = 1.0;
        double r1497629 = B;
        double r1497630 = tan(r1497629);
        double r1497631 = r1497628 / r1497630;
        double r1497632 = r1497627 * r1497631;
        double r1497633 = -r1497632;
        double r1497634 = F;
        double r1497635 = sin(r1497629);
        double r1497636 = r1497634 / r1497635;
        double r1497637 = r1497634 * r1497634;
        double r1497638 = 2.0;
        double r1497639 = r1497637 + r1497638;
        double r1497640 = r1497638 * r1497627;
        double r1497641 = r1497639 + r1497640;
        double r1497642 = r1497628 / r1497638;
        double r1497643 = -r1497642;
        double r1497644 = pow(r1497641, r1497643);
        double r1497645 = r1497636 * r1497644;
        double r1497646 = r1497633 + r1497645;
        return r1497646;
}
↓
double f(double F, double B, double x) {
        double r1497647 = F;
        double r1497648 = -2.034324775890907e+59;
        bool r1497649 = r1497647 <= r1497648;
        double r1497650 = 1.0;
        double r1497651 = B;
        double r1497652 = sin(r1497651);
        double r1497653 = r1497652 / r1497647;
        double r1497654 = -r1497653;
        double r1497655 = x;
        double r1497656 = r1497655 / r1497647;
        double r1497657 = r1497654 * r1497656;
        double r1497658 = r1497657 - r1497652;
        double r1497659 = r1497650 / r1497658;
        double r1497660 = tan(r1497651);
        double r1497661 = r1497655 / r1497660;
        double r1497662 = r1497659 - r1497661;
        double r1497663 = 107303683.05601409;
        bool r1497664 = r1497647 <= r1497663;
        double r1497665 = r1497647 * r1497647;
        double r1497666 = 2.0;
        double r1497667 = r1497665 + r1497666;
        double r1497668 = r1497655 * r1497666;
        double r1497669 = r1497667 + r1497668;
        double r1497670 = -0.5;
        double r1497671 = pow(r1497669, r1497670);
        double r1497672 = r1497671 / r1497653;
        double r1497673 = r1497672 - r1497661;
        double r1497674 = r1497655 / r1497665;
        double r1497675 = r1497674 * r1497652;
        double r1497676 = r1497652 + r1497675;
        double r1497677 = r1497650 / r1497676;
        double r1497678 = r1497677 - r1497661;
        double r1497679 = r1497664 ? r1497673 : r1497678;
        double r1497680 = r1497649 ? r1497662 : r1497679;
        return r1497680;
}

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 < -2.034324775890907e+59

    1. Initial program 27.6

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

    2. Simplified21.8

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

    4. Applied clear-num21.8

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

    5. Taylor expanded around -inf 0.2

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

    6. Simplified0.2

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

    if -2.034324775890907e+59 < F < 107303683.05601409

    1. Initial program 0.6

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

    2. Simplified0.3

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

    4. Applied associate-/l*0.5

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

    if 107303683.05601409 < F

    1. Initial program 26.0

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

    2. Simplified20.1

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

    4. Applied clear-num20.1

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -2.034324775890907 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\left(-\frac{\sin B}{F}\right) \cdot \frac{x}{F} - \sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 107303683.05601409:\\ \;\;\;\;\frac{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B + \frac{x}{F \cdot F} \cdot \sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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))))))