Average Error: 13.6 → 0.3
Time: 12.4s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -2.393293982823570639154866165724748883128 \cdot 10^{51}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{\frac{1}{F}}{F} - 1\right) \cdot \frac{1}{\sin B}\\ \mathbf{elif}\;F \le 16347121045823049728:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 - \frac{\frac{1}{F}}{F}\right) \cdot \frac{1}{\sin B}\\ \end{array}\]
\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 -2.393293982823570639154866165724748883128 \cdot 10^{51}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{\frac{1}{F}}{F} - 1\right) \cdot \frac{1}{\sin B}\\

\mathbf{elif}\;F \le 16347121045823049728:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \frac{1}{\sin B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r41318 = x;
        double r41319 = 1.0;
        double r41320 = B;
        double r41321 = tan(r41320);
        double r41322 = r41319 / r41321;
        double r41323 = r41318 * r41322;
        double r41324 = -r41323;
        double r41325 = F;
        double r41326 = sin(r41320);
        double r41327 = r41325 / r41326;
        double r41328 = r41325 * r41325;
        double r41329 = 2.0;
        double r41330 = r41328 + r41329;
        double r41331 = r41329 * r41318;
        double r41332 = r41330 + r41331;
        double r41333 = r41319 / r41329;
        double r41334 = -r41333;
        double r41335 = pow(r41332, r41334);
        double r41336 = r41327 * r41335;
        double r41337 = r41324 + r41336;
        return r41337;
}
↓
double f(double F, double B, double x) {
        double r41338 = F;
        double r41339 = -2.3932939828235706e+51;
        bool r41340 = r41338 <= r41339;
        double r41341 = x;
        double r41342 = 1.0;
        double r41343 = r41341 * r41342;
        double r41344 = B;
        double r41345 = tan(r41344);
        double r41346 = r41343 / r41345;
        double r41347 = -r41346;
        double r41348 = r41342 / r41338;
        double r41349 = r41348 / r41338;
        double r41350 = 1.0;
        double r41351 = r41349 - r41350;
        double r41352 = sin(r41344);
        double r41353 = r41350 / r41352;
        double r41354 = r41351 * r41353;
        double r41355 = r41347 + r41354;
        double r41356 = 1.634712104582305e+19;
        bool r41357 = r41338 <= r41356;
        double r41358 = cos(r41344);
        double r41359 = r41341 * r41358;
        double r41360 = r41359 / r41352;
        double r41361 = r41342 * r41360;
        double r41362 = -r41361;
        double r41363 = r41338 * r41338;
        double r41364 = 2.0;
        double r41365 = r41363 + r41364;
        double r41366 = r41364 * r41341;
        double r41367 = r41365 + r41366;
        double r41368 = r41342 / r41364;
        double r41369 = -r41368;
        double r41370 = pow(r41367, r41369);
        double r41371 = r41338 * r41370;
        double r41372 = r41371 * r41353;
        double r41373 = r41362 + r41372;
        double r41374 = r41350 - r41349;
        double r41375 = r41374 * r41353;
        double r41376 = r41347 + r41375;
        double r41377 = r41357 ? r41373 : r41376;
        double r41378 = r41340 ? r41355 : r41377;
        return r41378;
}

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 < -2.3932939828235706e+51

    1. Initial program 28.1

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

    3. Applied associate-*l/21.4

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

    5. Applied associate-*r/21.4

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

    7. Applied div-inv21.4

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

    8. Taylor expanded around -inf 0.2

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

    9. Simplified0.2

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

    if -2.3932939828235706e+51 < F < 1.634712104582305e+19

    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. Using strategy rm

    3. Applied associate-*l/0.4

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

    5. Applied associate-*r/0.3

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

    7. Applied div-inv0.3

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

    8. Taylor expanded around inf 0.4

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

    if 1.634712104582305e+19 < F

    1. Initial program 26.6

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

    3. Applied associate-*l/19.9

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

    5. Applied associate-*r/19.8

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

    7. Applied div-inv19.9

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

    8. Taylor expanded around inf 0.1

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

    9. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -2.393293982823570639154866165724748883128 \cdot 10^{51}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{\frac{1}{F}}{F} - 1\right) \cdot \frac{1}{\sin B}\\ \mathbf{elif}\;F \le 16347121045823049728:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 - \frac{\frac{1}{F}}{F}\right) \cdot \frac{1}{\sin B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))))))