Average Error: 13.8 → 0.3
Time: 23.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -13626.80779913700462202541530132293701172:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 4.348176063609124675224393286043778061867:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -13626.80779913700462202541530132293701172:\\
\;\;\;\;\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 4.348176063609124675224393286043778061867:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(-\frac{x \cdot 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 r47401 = x;
        double r47402 = 1.0;
        double r47403 = B;
        double r47404 = tan(r47403);
        double r47405 = r47402 / r47404;
        double r47406 = r47401 * r47405;
        double r47407 = -r47406;
        double r47408 = F;
        double r47409 = sin(r47403);
        double r47410 = r47408 / r47409;
        double r47411 = r47408 * r47408;
        double r47412 = 2.0;
        double r47413 = r47411 + r47412;
        double r47414 = r47412 * r47401;
        double r47415 = r47413 + r47414;
        double r47416 = r47402 / r47412;
        double r47417 = -r47416;
        double r47418 = pow(r47415, r47417);
        double r47419 = r47410 * r47418;
        double r47420 = r47407 + r47419;
        return r47420;
}
↓
double f(double F, double B, double x) {
        double r47421 = F;
        double r47422 = -13626.807799137005;
        bool r47423 = r47421 <= r47422;
        double r47424 = x;
        double r47425 = 1.0;
        double r47426 = r47424 * r47425;
        double r47427 = B;
        double r47428 = tan(r47427);
        double r47429 = r47426 / r47428;
        double r47430 = -r47429;
        double r47431 = 1.0;
        double r47432 = sin(r47427);
        double r47433 = 2.0;
        double r47434 = pow(r47421, r47433);
        double r47435 = r47432 * r47434;
        double r47436 = r47431 / r47435;
        double r47437 = r47425 * r47436;
        double r47438 = r47431 / r47432;
        double r47439 = r47437 - r47438;
        double r47440 = r47430 + r47439;
        double r47441 = 4.348176063609125;
        bool r47442 = r47421 <= r47441;
        double r47443 = cos(r47427);
        double r47444 = r47424 * r47443;
        double r47445 = r47444 / r47432;
        double r47446 = r47425 * r47445;
        double r47447 = -r47446;
        double r47448 = r47421 / r47432;
        double r47449 = r47421 * r47421;
        double r47450 = 2.0;
        double r47451 = r47449 + r47450;
        double r47452 = r47450 * r47424;
        double r47453 = r47451 + r47452;
        double r47454 = r47425 / r47450;
        double r47455 = -r47454;
        double r47456 = pow(r47453, r47455);
        double r47457 = r47448 * r47456;
        double r47458 = r47447 + r47457;
        double r47459 = r47438 - r47437;
        double r47460 = r47430 + r47459;
        double r47461 = r47442 ? r47458 : r47460;
        double r47462 = r47423 ? r47440 : r47461;
        return r47462;
}

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 < -13626.807799137005

    1. Initial program 25.8

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

    3. Applied pow-neg25.8

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

    4. Applied frac-times20.6

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

    5. Simplified20.6

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

    7. Applied associate-*r/20.5

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

    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 -13626.807799137005 < F < 4.348176063609125

    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. Taylor expanded around inf 0.3

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

    if 4.348176063609125 < F

    1. Initial program 23.8

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

    3. Applied pow-neg23.8

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

    4. Applied frac-times18.1

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

    5. Simplified18.1

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

    7. Applied associate-*r/18.1

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

    8. Taylor expanded around inf 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -13626.80779913700462202541530132293701172:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 4.348176063609124675224393286043778061867:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(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))))))