Average Error: 13.8 → 0.5
Time: 12.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -5.432067134143772600844375822214599802135 \cdot 10^{158}:\\ \;\;\;\;\left(-\left(x \cdot \frac{1}{\sin B}\right) \cdot \cos B\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 2.629705977215692284980606610579867920392 \cdot 10^{95}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\sin B} \cdot \cos B\right) + \frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -5.432067134143772600844375822214599802135 \cdot 10^{158}:\\
\;\;\;\;\left(-\left(x \cdot \frac{1}{\sin B}\right) \cdot \cos B\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\

\mathbf{elif}\;F \le 2.629705977215692284980606610579867920392 \cdot 10^{95}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\sin B} \cdot \cos B\right) + \frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r41279 = x;
        double r41280 = 1.0;
        double r41281 = B;
        double r41282 = tan(r41281);
        double r41283 = r41280 / r41282;
        double r41284 = r41279 * r41283;
        double r41285 = -r41284;
        double r41286 = F;
        double r41287 = sin(r41281);
        double r41288 = r41286 / r41287;
        double r41289 = r41286 * r41286;
        double r41290 = 2.0;
        double r41291 = r41289 + r41290;
        double r41292 = r41290 * r41279;
        double r41293 = r41291 + r41292;
        double r41294 = r41280 / r41290;
        double r41295 = -r41294;
        double r41296 = pow(r41293, r41295);
        double r41297 = r41288 * r41296;
        double r41298 = r41285 + r41297;
        return r41298;
}
↓
double f(double F, double B, double x) {
        double r41299 = F;
        double r41300 = -5.432067134143773e+158;
        bool r41301 = r41299 <= r41300;
        double r41302 = x;
        double r41303 = 1.0;
        double r41304 = B;
        double r41305 = sin(r41304);
        double r41306 = r41303 / r41305;
        double r41307 = r41302 * r41306;
        double r41308 = cos(r41304);
        double r41309 = r41307 * r41308;
        double r41310 = -r41309;
        double r41311 = 1.0;
        double r41312 = 2.0;
        double r41313 = pow(r41299, r41312);
        double r41314 = r41305 * r41313;
        double r41315 = r41311 / r41314;
        double r41316 = r41303 * r41315;
        double r41317 = r41311 / r41305;
        double r41318 = r41316 - r41317;
        double r41319 = r41310 + r41318;
        double r41320 = 2.6297059772156923e+95;
        bool r41321 = r41299 <= r41320;
        double r41322 = r41302 * r41303;
        double r41323 = r41322 / r41305;
        double r41324 = r41323 * r41308;
        double r41325 = -r41324;
        double r41326 = r41299 * r41299;
        double r41327 = 2.0;
        double r41328 = r41326 + r41327;
        double r41329 = r41327 * r41302;
        double r41330 = r41328 + r41329;
        double r41331 = r41303 / r41327;
        double r41332 = -r41331;
        double r41333 = pow(r41330, r41332);
        double r41334 = r41299 * r41333;
        double r41335 = r41334 / r41305;
        double r41336 = r41325 + r41335;
        double r41337 = tan(r41304);
        double r41338 = r41303 / r41337;
        double r41339 = r41302 * r41338;
        double r41340 = -r41339;
        double r41341 = r41317 - r41316;
        double r41342 = r41340 + r41341;
        double r41343 = r41321 ? r41336 : r41342;
        double r41344 = r41301 ? r41319 : r41343;
        return r41344;
}

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 < -5.432067134143773e+158

    1. Initial program 40.9

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

    3. Applied associate-*l/35.5

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

    5. Applied tan-quot35.5

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

    6. Applied associate-/r/35.5

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

    7. Applied associate-*r*35.5

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

    8. Taylor expanded around -inf 0.2

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

    if -5.432067134143773e+158 < F < 2.6297059772156923e+95

    1. Initial program 1.8

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

    3. Applied associate-*l/0.6

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

    5. Applied tan-quot0.6

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

    6. Applied associate-/r/0.6

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

    7. Applied associate-*r*0.6

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

    9. Applied associate-*r/0.6

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

    if 2.6297059772156923e+95 < F

    1. Initial program 34.9

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

    2. Taylor expanded around inf 0.2

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -5.432067134143772600844375822214599802135 \cdot 10^{158}:\\ \;\;\;\;\left(-\left(x \cdot \frac{1}{\sin B}\right) \cdot \cos B\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 2.629705977215692284980606610579867920392 \cdot 10^{95}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\sin B} \cdot \cos B\right) + \frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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))))))