Average Error: 13.3 → 0.2
Time: 39.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -3.0687291899060427 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 33330.37501774117:\\ \;\;\;\;\frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}} \cdot {\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -3.0687291899060427 \cdot 10^{+28}:\\
\;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \le 33330.37501774117:\\
\;\;\;\;\frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}} \cdot {\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r1216365 = x;
        double r1216366 = 1.0;
        double r1216367 = B;
        double r1216368 = tan(r1216367);
        double r1216369 = r1216366 / r1216368;
        double r1216370 = r1216365 * r1216369;
        double r1216371 = -r1216370;
        double r1216372 = F;
        double r1216373 = sin(r1216367);
        double r1216374 = r1216372 / r1216373;
        double r1216375 = r1216372 * r1216372;
        double r1216376 = 2.0;
        double r1216377 = r1216375 + r1216376;
        double r1216378 = r1216376 * r1216365;
        double r1216379 = r1216377 + r1216378;
        double r1216380 = r1216366 / r1216376;
        double r1216381 = -r1216380;
        double r1216382 = pow(r1216379, r1216381);
        double r1216383 = r1216374 * r1216382;
        double r1216384 = r1216371 + r1216383;
        return r1216384;
}
↓
double f(double F, double B, double x) {
        double r1216385 = F;
        double r1216386 = -3.0687291899060427e+28;
        bool r1216387 = r1216385 <= r1216386;
        double r1216388 = 1.0;
        double r1216389 = r1216385 * r1216385;
        double r1216390 = r1216388 / r1216389;
        double r1216391 = r1216390 - r1216388;
        double r1216392 = B;
        double r1216393 = sin(r1216392);
        double r1216394 = r1216391 / r1216393;
        double r1216395 = x;
        double r1216396 = tan(r1216392);
        double r1216397 = r1216395 / r1216396;
        double r1216398 = r1216394 - r1216397;
        double r1216399 = 33330.37501774117;
        bool r1216400 = r1216385 <= r1216399;
        double r1216401 = 2.0;
        double r1216402 = r1216401 * r1216395;
        double r1216403 = r1216402 + r1216389;
        double r1216404 = r1216403 + r1216401;
        double r1216405 = -0.25;
        double r1216406 = pow(r1216404, r1216405);
        double r1216407 = r1216406 * r1216406;
        double r1216408 = r1216393 / r1216385;
        double r1216409 = r1216407 / r1216408;
        double r1216410 = r1216409 - r1216397;
        double r1216411 = r1216388 - r1216390;
        double r1216412 = r1216411 / r1216393;
        double r1216413 = r1216412 - r1216397;
        double r1216414 = r1216400 ? r1216410 : r1216413;
        double r1216415 = r1216387 ? r1216398 : r1216414;
        return r1216415;
}

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 < -3.0687291899060427e+28

    1. Initial program 25.4

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

    2. Simplified24.6

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

    4. Applied div-inv24.6

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

    5. Applied *-un-lft-identity24.6

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

    6. Applied times-frac19.9

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

    7. Simplified19.9

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

    9. Applied associate-*l/19.9

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

    10. Simplified19.9

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

    11. Taylor expanded around -inf 0.2

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

    12. Simplified0.2

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

    if -3.0687291899060427e+28 < F < 33330.37501774117

    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

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

    4. Applied sqr-pow0.4

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

    if 33330.37501774117 < F

    1. Initial program 24.9

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

    2. Simplified24.4

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

    4. Applied div-inv24.4

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

    5. Applied *-un-lft-identity24.4

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

    6. Applied times-frac19.6

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

    7. Simplified19.6

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

    9. Applied associate-*l/19.6

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

    10. Simplified19.6

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

    11. Taylor expanded around inf 0.1

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

    12. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -3.0687291899060427 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 33330.37501774117:\\ \;\;\;\;\frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}} \cdot {\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{4}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(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))))))