Average Error: 14.4 → 0.4
Time: 1.3m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -306696806889543145500357201273043288064:\\ \;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} - \frac{1}{\sin B}\right) - \frac{x}{\tan B} \cdot 1\\ \mathbf{elif}\;F \le 3.830047883484379668376114121545286779086 \cdot 10^{78}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(x, 2, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x}{\tan B} \cdot 1\\ \end{array}\]
\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 -306696806889543145500357201273043288064:\\
\;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} - \frac{1}{\sin B}\right) - \frac{x}{\tan B} \cdot 1\\

\mathbf{elif}\;F \le 3.830047883484379668376114121545286779086 \cdot 10^{78}:\\
\;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(x, 2, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B} \cdot 1\\

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

\end{array}
double f(double F, double B, double x) {
        double r2107438 = x;
        double r2107439 = 1.0;
        double r2107440 = B;
        double r2107441 = tan(r2107440);
        double r2107442 = r2107439 / r2107441;
        double r2107443 = r2107438 * r2107442;
        double r2107444 = -r2107443;
        double r2107445 = F;
        double r2107446 = sin(r2107440);
        double r2107447 = r2107445 / r2107446;
        double r2107448 = r2107445 * r2107445;
        double r2107449 = 2.0;
        double r2107450 = r2107448 + r2107449;
        double r2107451 = r2107449 * r2107438;
        double r2107452 = r2107450 + r2107451;
        double r2107453 = r2107439 / r2107449;
        double r2107454 = -r2107453;
        double r2107455 = pow(r2107452, r2107454);
        double r2107456 = r2107447 * r2107455;
        double r2107457 = r2107444 + r2107456;
        return r2107457;
}
↓
double f(double F, double B, double x) {
        double r2107458 = F;
        double r2107459 = -3.0669680688954315e+38;
        bool r2107460 = r2107458 <= r2107459;
        double r2107461 = 1.0;
        double r2107462 = r2107458 * r2107458;
        double r2107463 = r2107461 / r2107462;
        double r2107464 = B;
        double r2107465 = sin(r2107464);
        double r2107466 = r2107463 / r2107465;
        double r2107467 = 1.0;
        double r2107468 = r2107467 / r2107465;
        double r2107469 = r2107466 - r2107468;
        double r2107470 = x;
        double r2107471 = tan(r2107464);
        double r2107472 = r2107470 / r2107471;
        double r2107473 = r2107472 * r2107461;
        double r2107474 = r2107469 - r2107473;
        double r2107475 = 3.83004788348438e+78;
        bool r2107476 = r2107458 <= r2107475;
        double r2107477 = r2107458 / r2107465;
        double r2107478 = 2.0;
        double r2107479 = fma(r2107470, r2107478, r2107478);
        double r2107480 = fma(r2107458, r2107458, r2107479);
        double r2107481 = r2107461 / r2107478;
        double r2107482 = pow(r2107480, r2107481);
        double r2107483 = r2107477 / r2107482;
        double r2107484 = r2107483 - r2107473;
        double r2107485 = r2107468 - r2107466;
        double r2107486 = r2107485 - r2107473;
        double r2107487 = r2107476 ? r2107484 : r2107486;
        double r2107488 = r2107460 ? r2107474 : r2107487;
        return r2107488;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -3.0669680688954315e+38

    1. Initial program 28.5

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

    2. Simplified21.8

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

    4. Applied div-inv21.8

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

    5. Applied associate-*l*21.8

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

    6. Simplified21.8

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

    7. Taylor expanded around -inf 0.2

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

    8. Simplified0.2

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

    if -3.0669680688954315e+38 < F < 3.83004788348438e+78

    1. Initial program 0.7

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

    2. Simplified0.4

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

    4. Applied div-inv0.4

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

    5. Applied associate-*l*0.4

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

    6. Simplified0.3

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

    8. Applied pow-neg0.3

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

    9. Applied associate-/r/0.3

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

    10. Applied associate-/r*0.6

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

    11. Simplified0.6

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

    if 3.83004788348438e+78 < F

    1. Initial program 32.0

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

    2. Simplified25.1

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

    4. Applied div-inv25.1

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

    5. Applied associate-*l*25.1

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

    6. Simplified25.1

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

    7. Taylor expanded around inf 0.1

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

    8. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -306696806889543145500357201273043288064:\\ \;\;\;\;\left(\frac{\frac{1}{F \cdot F}}{\sin B} - \frac{1}{\sin B}\right) - \frac{x}{\tan B} \cdot 1\\ \mathbf{elif}\;F \le 3.830047883484379668376114121545286779086 \cdot 10^{78}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(x, 2, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}} - \frac{x}{\tan B} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x}{\tan B} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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))))))