Average Error: 13.4 → 0.3
Time: 14.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -2230687376.113047122955322265625:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 96.48230808888776266485365340486168861389:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + F \cdot \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -2230687376.113047122955322265625:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\

\mathbf{elif}\;F \le 96.48230808888776266485365340486168861389:\\
\;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + F \cdot \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\sin B}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(-x \cdot \frac{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 r52476 = x;
        double r52477 = 1.0;
        double r52478 = B;
        double r52479 = tan(r52478);
        double r52480 = r52477 / r52479;
        double r52481 = r52476 * r52480;
        double r52482 = -r52481;
        double r52483 = F;
        double r52484 = sin(r52478);
        double r52485 = r52483 / r52484;
        double r52486 = r52483 * r52483;
        double r52487 = 2.0;
        double r52488 = r52486 + r52487;
        double r52489 = r52487 * r52476;
        double r52490 = r52488 + r52489;
        double r52491 = r52477 / r52487;
        double r52492 = -r52491;
        double r52493 = pow(r52490, r52492);
        double r52494 = r52485 * r52493;
        double r52495 = r52482 + r52494;
        return r52495;
}
↓
double f(double F, double B, double x) {
        double r52496 = F;
        double r52497 = -2230687376.113047;
        bool r52498 = r52496 <= r52497;
        double r52499 = x;
        double r52500 = 1.0;
        double r52501 = r52499 * r52500;
        double r52502 = B;
        double r52503 = tan(r52502);
        double r52504 = r52501 / r52503;
        double r52505 = -r52504;
        double r52506 = 1.0;
        double r52507 = sin(r52502);
        double r52508 = 2.0;
        double r52509 = pow(r52496, r52508);
        double r52510 = r52507 * r52509;
        double r52511 = r52506 / r52510;
        double r52512 = r52500 * r52511;
        double r52513 = r52506 / r52507;
        double r52514 = r52512 - r52513;
        double r52515 = r52505 + r52514;
        double r52516 = 96.48230808888776;
        bool r52517 = r52496 <= r52516;
        double r52518 = r52500 / r52503;
        double r52519 = r52499 * r52518;
        double r52520 = -r52519;
        double r52521 = r52496 * r52496;
        double r52522 = 2.0;
        double r52523 = r52521 + r52522;
        double r52524 = r52522 * r52499;
        double r52525 = r52523 + r52524;
        double r52526 = r52500 / r52522;
        double r52527 = -r52526;
        double r52528 = pow(r52525, r52527);
        double r52529 = r52528 * r52513;
        double r52530 = r52496 * r52529;
        double r52531 = r52520 + r52530;
        double r52532 = r52513 - r52512;
        double r52533 = r52520 + r52532;
        double r52534 = r52517 ? r52531 : r52533;
        double r52535 = r52498 ? r52515 : r52534;
        return r52535;
}

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 < -2230687376.113047

    1. Initial program 24.3

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

    3. Applied div-inv24.3

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

    4. Applied associate-*l*18.9

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

    5. Simplified19.0

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

    7. Applied associate-*r/18.9

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

    8. Taylor expanded around -inf 0.2

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

    if -2230687376.113047 < F < 96.48230808888776

    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. Using strategy rm

    3. Applied div-inv0.4

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

    4. Applied associate-*l*0.4

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

    5. Simplified0.4

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

    7. Applied div-inv0.4

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

    if 96.48230808888776 < F

    1. Initial program 25.3

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

    2. Taylor expanded around inf 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -2230687376.113047122955322265625:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 96.48230808888776266485365340486168861389:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + F \cdot \left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))))