Average Error: 13.9 → 0.3
Time: 42.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -2.500335819505326495098846233879206770216 \cdot 10^{53}:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 196580.8150627871218603104352951049804688:\\ \;\;\;\;F \cdot \frac{{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \cos B \cdot \frac{x \cdot 1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]
\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 -2.500335819505326495098846233879206770216 \cdot 10^{53}:\\
\;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 196580.8150627871218603104352951049804688:\\
\;\;\;\;F \cdot \frac{{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \cos B \cdot \frac{x \cdot 1}{\sin B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r2293322 = x;
        double r2293323 = 1.0;
        double r2293324 = B;
        double r2293325 = tan(r2293324);
        double r2293326 = r2293323 / r2293325;
        double r2293327 = r2293322 * r2293326;
        double r2293328 = -r2293327;
        double r2293329 = F;
        double r2293330 = sin(r2293324);
        double r2293331 = r2293329 / r2293330;
        double r2293332 = r2293329 * r2293329;
        double r2293333 = 2.0;
        double r2293334 = r2293332 + r2293333;
        double r2293335 = r2293333 * r2293322;
        double r2293336 = r2293334 + r2293335;
        double r2293337 = r2293323 / r2293333;
        double r2293338 = -r2293337;
        double r2293339 = pow(r2293336, r2293338);
        double r2293340 = r2293331 * r2293339;
        double r2293341 = r2293328 + r2293340;
        return r2293341;
}
↓
double f(double F, double B, double x) {
        double r2293342 = F;
        double r2293343 = -2.5003358195053265e+53;
        bool r2293344 = r2293342 <= r2293343;
        double r2293345 = 1.0;
        double r2293346 = B;
        double r2293347 = sin(r2293346);
        double r2293348 = r2293345 / r2293347;
        double r2293349 = r2293342 * r2293342;
        double r2293350 = r2293348 / r2293349;
        double r2293351 = -1.0;
        double r2293352 = r2293351 / r2293347;
        double r2293353 = r2293350 + r2293352;
        double r2293354 = x;
        double r2293355 = r2293354 * r2293345;
        double r2293356 = tan(r2293346);
        double r2293357 = r2293355 / r2293356;
        double r2293358 = r2293353 - r2293357;
        double r2293359 = 196580.81506278712;
        bool r2293360 = r2293342 <= r2293359;
        double r2293361 = 2.0;
        double r2293362 = r2293361 + r2293349;
        double r2293363 = r2293354 * r2293361;
        double r2293364 = r2293362 + r2293363;
        double r2293365 = r2293345 / r2293361;
        double r2293366 = -r2293365;
        double r2293367 = pow(r2293364, r2293366);
        double r2293368 = r2293367 / r2293347;
        double r2293369 = r2293342 * r2293368;
        double r2293370 = cos(r2293346);
        double r2293371 = r2293355 / r2293347;
        double r2293372 = r2293370 * r2293371;
        double r2293373 = r2293369 - r2293372;
        double r2293374 = 1.0;
        double r2293375 = r2293374 / r2293347;
        double r2293376 = r2293375 - r2293350;
        double r2293377 = r2293376 - r2293357;
        double r2293378 = r2293360 ? r2293373 : r2293377;
        double r2293379 = r2293344 ? r2293358 : r2293378;
        return r2293379;
}

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 < -2.5003358195053265e+53

    1. Initial program 29.4

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

    2. Simplified28.5

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

    4. Applied associate-/r/22.1

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

    6. Applied associate-*l/22.0

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

    7. Taylor expanded around -inf 0.2

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

    8. Simplified0.2

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

    if -2.5003358195053265e+53 < F < 196580.81506278712

    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. Simplified0.5

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

    4. Applied associate-/r/0.4

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

    6. Applied associate-*l/0.3

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

    8. Applied tan-quot0.3

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

    9. Applied associate-/r/0.3

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

    if 196580.81506278712 < F

    1. Initial program 24.7

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

    2. Simplified24.0

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

    4. Applied associate-/r/18.5

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

    6. Applied associate-*l/18.4

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

    7. Taylor expanded around inf 0.2

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

    8. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -2.500335819505326495098846233879206770216 \cdot 10^{53}:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 196580.8150627871218603104352951049804688:\\ \;\;\;\;F \cdot \frac{{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \cos B \cdot \frac{x \cdot 1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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))))))