Average Error: 13.2 → 0.2
Time: 33.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -256364438.11374387:\\ \;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 53809784.808465004:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\frac{-1}{2}} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array}\]
\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 -256364438.11374387:\\
\;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\

\mathbf{elif}\;F \le 53809784.808465004:\\
\;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\frac{-1}{2}} - \frac{x \cdot \cos B}{\sin B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\

\end{array}
double f(double F, double B, double x) {
        double r888428 = x;
        double r888429 = 1.0;
        double r888430 = B;
        double r888431 = tan(r888430);
        double r888432 = r888429 / r888431;
        double r888433 = r888428 * r888432;
        double r888434 = -r888433;
        double r888435 = F;
        double r888436 = sin(r888430);
        double r888437 = r888435 / r888436;
        double r888438 = r888435 * r888435;
        double r888439 = 2.0;
        double r888440 = r888438 + r888439;
        double r888441 = r888439 * r888428;
        double r888442 = r888440 + r888441;
        double r888443 = r888429 / r888439;
        double r888444 = -r888443;
        double r888445 = pow(r888442, r888444);
        double r888446 = r888437 * r888445;
        double r888447 = r888434 + r888446;
        return r888447;
}
↓
double f(double F, double B, double x) {
        double r888448 = F;
        double r888449 = -256364438.11374387;
        bool r888450 = r888448 <= r888449;
        double r888451 = 1.0;
        double r888452 = r888448 * r888448;
        double r888453 = r888451 / r888452;
        double r888454 = B;
        double r888455 = sin(r888454);
        double r888456 = r888453 / r888455;
        double r888457 = r888451 / r888455;
        double r888458 = r888456 - r888457;
        double r888459 = x;
        double r888460 = tan(r888454);
        double r888461 = r888459 / r888460;
        double r888462 = r888458 - r888461;
        double r888463 = 53809784.808465004;
        bool r888464 = r888448 <= r888463;
        double r888465 = r888448 / r888455;
        double r888466 = 2.0;
        double r888467 = r888459 * r888466;
        double r888468 = r888466 + r888452;
        double r888469 = r888467 + r888468;
        double r888470 = -0.5;
        double r888471 = pow(r888469, r888470);
        double r888472 = r888465 * r888471;
        double r888473 = cos(r888454);
        double r888474 = r888459 * r888473;
        double r888475 = r888474 / r888455;
        double r888476 = r888472 - r888475;
        double r888477 = r888451 / r888448;
        double r888478 = r888477 / r888448;
        double r888479 = r888451 - r888478;
        double r888480 = r888479 / r888455;
        double r888481 = r888459 / r888455;
        double r888482 = r888473 * r888481;
        double r888483 = r888480 - r888482;
        double r888484 = r888464 ? r888476 : r888483;
        double r888485 = r888450 ? r888462 : r888484;
        return r888485;
}

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 < -256364438.11374387

    1. Initial program 23.7

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

    2. Simplified23.6

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

    4. Applied associate-*r/18.9

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

    5. Taylor expanded around -inf 0.1

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

    6. Simplified0.1

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

    if -256364438.11374387 < F < 53809784.808465004

    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. Simplified0.3

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

    3. Taylor expanded around -inf 0.3

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

    if 53809784.808465004 < F

    1. Initial program 25.2

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

    2. Simplified25.1

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

    4. Applied associate-*r/19.8

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

    6. Applied tan-quot19.8

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

    7. Applied associate-/r/19.8

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

    8. Taylor expanded around inf 0.2

      \[\leadsto \frac{\color{blue}{1 - \frac{1}{{F}^{2}}}}{\sin B} - \frac{x}{\sin B} \cdot \cos B\]

    9. Simplified0.2

      \[\leadsto \frac{\color{blue}{1 - \frac{\frac{1}{F}}{F}}}{\sin B} - \frac{x}{\sin B} \cdot \cos B\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -256364438.11374387:\\ \;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 53809784.808465004:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\frac{-1}{2}} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))