Average Error: 14.2 → 0.3
Time: 17.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -15429963473208182800:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 7.1670148200446107 \cdot 10^{107}:\\ \;\;\;\;F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array}\]
\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 -15429963473208182800:\\
\;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 7.1670148200446107 \cdot 10^{107}:\\
\;\;\;\;F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r48512 = x;
        double r48513 = 1.0;
        double r48514 = B;
        double r48515 = tan(r48514);
        double r48516 = r48513 / r48515;
        double r48517 = r48512 * r48516;
        double r48518 = -r48517;
        double r48519 = F;
        double r48520 = sin(r48514);
        double r48521 = r48519 / r48520;
        double r48522 = r48519 * r48519;
        double r48523 = 2.0;
        double r48524 = r48522 + r48523;
        double r48525 = r48523 * r48512;
        double r48526 = r48524 + r48525;
        double r48527 = r48513 / r48523;
        double r48528 = -r48527;
        double r48529 = pow(r48526, r48528);
        double r48530 = r48521 * r48529;
        double r48531 = r48518 + r48530;
        return r48531;
}
↓
double f(double F, double B, double x) {
        double r48532 = F;
        double r48533 = -1.5429963473208183e+19;
        bool r48534 = r48532 <= r48533;
        double r48535 = 1.0;
        double r48536 = B;
        double r48537 = sin(r48536);
        double r48538 = 2.0;
        double r48539 = pow(r48532, r48538);
        double r48540 = r48537 * r48539;
        double r48541 = r48535 / r48540;
        double r48542 = -1.0;
        double r48543 = r48542 / r48537;
        double r48544 = r48541 + r48543;
        double r48545 = x;
        double r48546 = r48545 * r48535;
        double r48547 = tan(r48536);
        double r48548 = r48546 / r48547;
        double r48549 = r48544 - r48548;
        double r48550 = 7.167014820044611e+107;
        bool r48551 = r48532 <= r48550;
        double r48552 = r48532 * r48532;
        double r48553 = 2.0;
        double r48554 = r48552 + r48553;
        double r48555 = r48553 * r48545;
        double r48556 = r48554 + r48555;
        double r48557 = r48535 / r48553;
        double r48558 = -r48557;
        double r48559 = pow(r48556, r48558);
        double r48560 = r48559 / r48537;
        double r48561 = r48532 * r48560;
        double r48562 = cos(r48536);
        double r48563 = r48545 * r48562;
        double r48564 = r48563 / r48537;
        double r48565 = r48535 * r48564;
        double r48566 = r48561 - r48565;
        double r48567 = 1.0;
        double r48568 = r48567 / r48537;
        double r48569 = r48568 - r48541;
        double r48570 = r48535 / r48547;
        double r48571 = r48545 * r48570;
        double r48572 = r48569 - r48571;
        double r48573 = r48551 ? r48566 : r48572;
        double r48574 = r48534 ? r48549 : r48573;
        return r48574;
}

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 < -1.5429963473208183e+19

    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. Simplified26.6

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

    4. Applied div-inv26.6

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

    5. Applied associate-*l*20.4

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

    6. Simplified20.5

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

    8. Applied associate-*r/20.4

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

    9. Taylor expanded around -inf 0.1

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

    10. Simplified0.1

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

    if -1.5429963473208183e+19 < F < 7.167014820044611e+107

    1. Initial program 1.0

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

    2. Simplified1.0

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

    4. Applied div-inv1.0

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

    5. Applied associate-*l*0.4

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

    6. Simplified0.4

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

    7. Taylor expanded around inf 0.3

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

    if 7.167014820044611e+107 < F

    1. Initial program 35.8

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

    2. Simplified35.8

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -15429963473208182800:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 7.1670148200446107 \cdot 10^{107}:\\ \;\;\;\;F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))))