Average Error: 14.2 → 0.2
Time: 10.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -25045099127.8391991:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 1584339.73845752398:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]
\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 -25045099127.8391991:\\
\;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 1584339.73845752398:\\
\;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\sin B} \cdot \cos B\\

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

\end{array}
double f(double F, double B, double x) {
        double r39357 = x;
        double r39358 = 1.0;
        double r39359 = B;
        double r39360 = tan(r39359);
        double r39361 = r39358 / r39360;
        double r39362 = r39357 * r39361;
        double r39363 = -r39362;
        double r39364 = F;
        double r39365 = sin(r39359);
        double r39366 = r39364 / r39365;
        double r39367 = r39364 * r39364;
        double r39368 = 2.0;
        double r39369 = r39367 + r39368;
        double r39370 = r39368 * r39357;
        double r39371 = r39369 + r39370;
        double r39372 = r39358 / r39368;
        double r39373 = -r39372;
        double r39374 = pow(r39371, r39373);
        double r39375 = r39366 * r39374;
        double r39376 = r39363 + r39375;
        return r39376;
}
↓
double f(double F, double B, double x) {
        double r39377 = F;
        double r39378 = -25045099127.8392;
        bool r39379 = r39377 <= r39378;
        double r39380 = 1.0;
        double r39381 = 1.0;
        double r39382 = B;
        double r39383 = sin(r39382);
        double r39384 = 2.0;
        double r39385 = pow(r39377, r39384);
        double r39386 = r39383 * r39385;
        double r39387 = r39381 / r39386;
        double r39388 = r39380 * r39387;
        double r39389 = r39381 / r39383;
        double r39390 = r39388 - r39389;
        double r39391 = x;
        double r39392 = r39391 * r39380;
        double r39393 = tan(r39382);
        double r39394 = r39392 / r39393;
        double r39395 = r39390 - r39394;
        double r39396 = 1584339.738457524;
        bool r39397 = r39377 <= r39396;
        double r39398 = r39377 / r39383;
        double r39399 = r39377 * r39377;
        double r39400 = 2.0;
        double r39401 = r39399 + r39400;
        double r39402 = r39400 * r39391;
        double r39403 = r39401 + r39402;
        double r39404 = r39380 / r39400;
        double r39405 = pow(r39403, r39404);
        double r39406 = r39398 / r39405;
        double r39407 = r39392 / r39383;
        double r39408 = cos(r39382);
        double r39409 = r39407 * r39408;
        double r39410 = r39406 - r39409;
        double r39411 = r39389 - r39388;
        double r39412 = r39411 - r39394;
        double r39413 = r39397 ? r39410 : r39412;
        double r39414 = r39379 ? r39395 : r39413;
        return r39414;
}

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 < -25045099127.8392

    1. Initial program 26.2

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

    2. Simplified26.2

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

    4. Applied associate-*r/26.1

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

    5. Taylor expanded around -inf 0.1

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

    if -25045099127.8392 < F < 1584339.738457524

    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. Simplified0.4

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

    4. Applied associate-*r/0.3

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

    6. Applied pow-neg0.3

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

    7. Applied un-div-inv0.3

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

    9. Applied tan-quot0.3

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

    10. Applied associate-/r/0.3

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

    if 1584339.738457524 < F

    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. Simplified26.4

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

    4. Applied associate-*r/26.3

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

    5. Taylor expanded around inf 0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -25045099127.8391991:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 1584339.73845752398:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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))))))