Average Error: 13.5 → 0.2
Time: 32.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -221911557169.175994873046875:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 65507857.242128990590572357177734375:\\ \;\;\;\;\frac{F \cdot {\left(\sqrt{F \cdot F + \left(x \cdot 2 + 2\right)}\right)}^{\left(-\frac{1}{2}\right)}}{{\left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]
\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 -221911557169.175994873046875:\\
\;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 65507857.242128990590572357177734375:\\
\;\;\;\;\frac{F \cdot {\left(\sqrt{F \cdot F + \left(x \cdot 2 + 2\right)}\right)}^{\left(-\frac{1}{2}\right)}}{{\left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r56364 = x;
        double r56365 = 1.0;
        double r56366 = B;
        double r56367 = tan(r56366);
        double r56368 = r56365 / r56367;
        double r56369 = r56364 * r56368;
        double r56370 = -r56369;
        double r56371 = F;
        double r56372 = sin(r56366);
        double r56373 = r56371 / r56372;
        double r56374 = r56371 * r56371;
        double r56375 = 2.0;
        double r56376 = r56374 + r56375;
        double r56377 = r56375 * r56364;
        double r56378 = r56376 + r56377;
        double r56379 = r56365 / r56375;
        double r56380 = -r56379;
        double r56381 = pow(r56378, r56380);
        double r56382 = r56373 * r56381;
        double r56383 = r56370 + r56382;
        return r56383;
}
↓
double f(double F, double B, double x) {
        double r56384 = F;
        double r56385 = -221911557169.176;
        bool r56386 = r56384 <= r56385;
        double r56387 = 1.0;
        double r56388 = B;
        double r56389 = sin(r56388);
        double r56390 = r56387 / r56389;
        double r56391 = r56384 * r56384;
        double r56392 = r56390 / r56391;
        double r56393 = 1.0;
        double r56394 = r56393 / r56389;
        double r56395 = r56392 - r56394;
        double r56396 = x;
        double r56397 = r56396 * r56387;
        double r56398 = tan(r56388);
        double r56399 = r56397 / r56398;
        double r56400 = r56395 - r56399;
        double r56401 = 65507857.24212899;
        bool r56402 = r56384 <= r56401;
        double r56403 = 2.0;
        double r56404 = r56396 * r56403;
        double r56405 = r56404 + r56403;
        double r56406 = r56391 + r56405;
        double r56407 = sqrt(r56406);
        double r56408 = r56387 / r56403;
        double r56409 = -r56408;
        double r56410 = pow(r56407, r56409);
        double r56411 = r56384 * r56410;
        double r56412 = r56391 + r56403;
        double r56413 = r56404 + r56412;
        double r56414 = sqrt(r56413);
        double r56415 = pow(r56414, r56408);
        double r56416 = r56415 * r56389;
        double r56417 = r56411 / r56416;
        double r56418 = r56417 - r56399;
        double r56419 = r56387 / r56391;
        double r56420 = r56419 / r56389;
        double r56421 = r56394 - r56420;
        double r56422 = r56421 - r56399;
        double r56423 = r56402 ? r56418 : r56422;
        double r56424 = r56386 ? r56400 : r56423;
        return r56424;
}

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 < -221911557169.176

    1. Initial program 25.0

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

    2. Simplified19.5

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

    if -221911557169.176 < F < 65507857.24212899

    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

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

    4. Applied add-sqr-sqrt0.3

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

    5. Applied unpow-prod-down0.3

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

    6. Applied *-un-lft-identity0.3

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

    7. Applied times-frac0.3

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

    8. Applied associate-/r*0.3

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

    9. Simplified0.3

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

    11. Applied pow-neg0.3

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

    12. Applied associate-/r/0.2

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

    13. Simplified0.2

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

    if 65507857.24212899 < F

    1. Initial program 25.5

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

    2. Simplified19.4

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

    4. Applied pow-neg19.4

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

    5. Applied associate-/r/19.4

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

    6. Applied *-un-lft-identity19.4

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

    7. Applied times-frac19.4

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

    8. Simplified19.4

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

    9. Simplified19.4

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

    11. Applied frac-times19.4

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

    12. Simplified19.4

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

    13. Simplified19.4

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

    14. Taylor expanded around inf 0.1

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

    15. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -221911557169.175994873046875:\\ \;\;\;\;\left(\frac{\frac{1}{\sin B}}{F \cdot F} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 65507857.242128990590572357177734375:\\ \;\;\;\;\frac{F \cdot {\left(\sqrt{F \cdot F + \left(x \cdot 2 + 2\right)}\right)}^{\left(-\frac{1}{2}\right)}}{{\left(\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))))))