Average Error: 14.4 → 0.2
Time: 38.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -61781114441641127683218472587214714361090000:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 3810689.7252205349504947662353515625:\\ \;\;\;\;\frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(1 - \frac{1}{F \cdot F}\right) - \frac{1}{\frac{\tan B}{x \cdot 1}}\\ \end{array}\]
\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 -61781114441641127683218472587214714361090000:\\
\;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 3810689.7252205349504947662353515625:\\
\;\;\;\;\frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r2393347 = x;
        double r2393348 = 1.0;
        double r2393349 = B;
        double r2393350 = tan(r2393349);
        double r2393351 = r2393348 / r2393350;
        double r2393352 = r2393347 * r2393351;
        double r2393353 = -r2393352;
        double r2393354 = F;
        double r2393355 = sin(r2393349);
        double r2393356 = r2393354 / r2393355;
        double r2393357 = r2393354 * r2393354;
        double r2393358 = 2.0;
        double r2393359 = r2393357 + r2393358;
        double r2393360 = r2393358 * r2393347;
        double r2393361 = r2393359 + r2393360;
        double r2393362 = r2393348 / r2393358;
        double r2393363 = -r2393362;
        double r2393364 = pow(r2393361, r2393363);
        double r2393365 = r2393356 * r2393364;
        double r2393366 = r2393353 + r2393365;
        return r2393366;
}
↓
double f(double F, double B, double x) {
        double r2393367 = F;
        double r2393368 = -6.178111444164113e+43;
        bool r2393369 = r2393367 <= r2393368;
        double r2393370 = 1.0;
        double r2393371 = B;
        double r2393372 = sin(r2393371);
        double r2393373 = r2393370 / r2393372;
        double r2393374 = r2393367 * r2393367;
        double r2393375 = r2393373 / r2393374;
        double r2393376 = 1.0;
        double r2393377 = r2393376 / r2393372;
        double r2393378 = r2393375 - r2393377;
        double r2393379 = x;
        double r2393380 = r2393379 * r2393370;
        double r2393381 = tan(r2393371);
        double r2393382 = r2393380 / r2393381;
        double r2393383 = r2393378 - r2393382;
        double r2393384 = 3810689.725220535;
        bool r2393385 = r2393367 <= r2393384;
        double r2393386 = 2.0;
        double r2393387 = r2393374 + r2393386;
        double r2393388 = r2393379 * r2393386;
        double r2393389 = r2393387 + r2393388;
        double r2393390 = r2393370 / r2393386;
        double r2393391 = pow(r2393389, r2393390);
        double r2393392 = r2393391 * r2393372;
        double r2393393 = r2393367 / r2393392;
        double r2393394 = r2393393 - r2393382;
        double r2393395 = r2393370 / r2393374;
        double r2393396 = r2393376 - r2393395;
        double r2393397 = r2393377 * r2393396;
        double r2393398 = r2393381 / r2393380;
        double r2393399 = r2393376 / r2393398;
        double r2393400 = r2393397 - r2393399;
        double r2393401 = r2393385 ? r2393394 : r2393400;
        double r2393402 = r2393369 ? r2393383 : r2393401;
        return r2393402;
}

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 < -6.178111444164113e+43

    1. Initial program 29.0

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

    2. Simplified22.3

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -6.178111444164113e+43 < F < 3810689.725220535

    1. Initial program 0.5

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

    2. Simplified0.3

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

    4. Applied pow-neg0.3

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

    5. Applied associate-/r/0.3

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

    6. Simplified0.3

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

    if 3810689.725220535 < F

    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. Simplified20.1

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

    4. Applied div-inv20.1

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

    5. Applied *-un-lft-identity20.1

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

    6. Applied times-frac20.0

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

    7. Simplified20.0

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

    9. Applied clear-num20.1

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

    10. Taylor expanded around inf 0.2

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

    11. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -61781114441641127683218472587214714361090000:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 3810689.7252205349504947662353515625:\\ \;\;\;\;\frac{F}{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(1 - \frac{1}{F \cdot F}\right) - \frac{1}{\frac{\tan B}{x \cdot 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))