Average Error: 13.4 → 0.7
Time: 5.2m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1.9108912601642475 \cdot 10^{+100}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 3.7275517711903683 \cdot 10^{+65}:\\ \;\;\;\;\frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x}{\tan B}\\ \end{array}\]
double f(double F, double B, double x) {
        double r24111346 = x;
        double r24111347 = 1.0;
        double r24111348 = B;
        double r24111349 = tan(r24111348);
        double r24111350 = r24111347 / r24111349;
        double r24111351 = r24111346 * r24111350;
        double r24111352 = -r24111351;
        double r24111353 = F;
        double r24111354 = sin(r24111348);
        double r24111355 = r24111353 / r24111354;
        double r24111356 = r24111353 * r24111353;
        double r24111357 = 2.0;
        double r24111358 = r24111356 + r24111357;
        double r24111359 = r24111357 * r24111346;
        double r24111360 = r24111358 + r24111359;
        double r24111361 = r24111347 / r24111357;
        double r24111362 = -r24111361;
        double r24111363 = pow(r24111360, r24111362);
        double r24111364 = r24111355 * r24111363;
        double r24111365 = r24111352 + r24111364;
        return r24111365;
}
↓
double f(double F, double B, double x) {
        double r24111366 = F;
        double r24111367 = -1.9108912601642475e+100;
        bool r24111368 = r24111366 <= r24111367;
        double r24111369 = 1.0;
        double r24111370 = B;
        double r24111371 = sin(r24111370);
        double r24111372 = r24111366 * r24111366;
        double r24111373 = r24111371 * r24111372;
        double r24111374 = r24111369 / r24111373;
        double r24111375 = r24111369 / r24111371;
        double r24111376 = r24111374 - r24111375;
        double r24111377 = x;
        double r24111378 = tan(r24111370);
        double r24111379 = r24111377 / r24111378;
        double r24111380 = r24111376 - r24111379;
        double r24111381 = 3.7275517711903683e+65;
        bool r24111382 = r24111366 <= r24111381;
        double r24111383 = 2.0;
        double r24111384 = fma(r24111366, r24111366, r24111383);
        double r24111385 = fma(r24111383, r24111377, r24111384);
        double r24111386 = -0.5;
        double r24111387 = pow(r24111385, r24111386);
        double r24111388 = r24111371 / r24111366;
        double r24111389 = r24111387 / r24111388;
        double r24111390 = r24111377 / r24111371;
        double r24111391 = cos(r24111370);
        double r24111392 = r24111390 * r24111391;
        double r24111393 = r24111389 - r24111392;
        double r24111394 = r24111375 / r24111372;
        double r24111395 = r24111375 - r24111394;
        double r24111396 = r24111395 - r24111379;
        double r24111397 = r24111382 ? r24111393 : r24111396;
        double r24111398 = r24111368 ? r24111380 : r24111397;
        return r24111398;
}


\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 -1.9108912601642475 \cdot 10^{+100}:\\
\;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\

\mathbf{elif}\;F \le 3.7275517711903683 \cdot 10^{+65}:\\
\;\;\;\;\frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\sin B} \cdot \cos B\\

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

\end{array}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -1.9108912601642475e+100

    1. Initial program 33.1

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

    2. Simplified31.8

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

    4. Applied add-cube-cbrt31.9

      \[\leadsto \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sin B}{\color{blue}{\left(\sqrt[3]{F} \cdot \sqrt[3]{F}\right) \cdot \sqrt[3]{F}}}} - \frac{x}{\tan B}\]

    5. Applied add-cube-cbrt31.9

      \[\leadsto \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\color{blue}{\left(\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}\right) \cdot \sqrt[3]{\sin B}}}{\left(\sqrt[3]{F} \cdot \sqrt[3]{F}\right) \cdot \sqrt[3]{F}}} - \frac{x}{\tan B}\]

    6. Applied times-frac31.9

      \[\leadsto \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\color{blue}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}}} - \frac{x}{\tan B}\]

    7. Applied *-un-lft-identity31.9

      \[\leadsto \frac{{\color{blue}{\left(1 \cdot (2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}} - \frac{x}{\tan B}\]

    8. Applied unpow-prod-down31.9

      \[\leadsto \frac{\color{blue}{{1}^{\frac{-1}{2}} \cdot {\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}} - \frac{x}{\tan B}\]

    9. Applied times-frac27.6

      \[\leadsto \color{blue}{\frac{{1}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}}} \cdot \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}}} - \frac{x}{\tan B}\]

    10. Simplified27.6

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{F}}{\sqrt[3]{\sin B}} \cdot \frac{\sqrt[3]{F}}{\sqrt[3]{\sin B}}\right)} \cdot \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}} - \frac{x}{\tan B}\]

    11. Taylor expanded around -inf 0.2

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

    12. Simplified0.2

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

    if -1.9108912601642475e+100 < F < 3.7275517711903683e+65

    1. Initial program 1.2

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

    2. Simplified1.0

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

    4. Applied tan-quot1.0

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

    5. Applied associate-/r/1.0

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

    if 3.7275517711903683e+65 < F

    1. Initial program 30.4

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

    2. Simplified29.5

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

    4. Applied add-cube-cbrt29.6

      \[\leadsto \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sin B}{\color{blue}{\left(\sqrt[3]{F} \cdot \sqrt[3]{F}\right) \cdot \sqrt[3]{F}}}} - \frac{x}{\tan B}\]

    5. Applied add-cube-cbrt29.7

      \[\leadsto \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\color{blue}{\left(\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}\right) \cdot \sqrt[3]{\sin B}}}{\left(\sqrt[3]{F} \cdot \sqrt[3]{F}\right) \cdot \sqrt[3]{F}}} - \frac{x}{\tan B}\]

    6. Applied times-frac29.7

      \[\leadsto \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\color{blue}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}}} - \frac{x}{\tan B}\]

    7. Applied *-un-lft-identity29.7

      \[\leadsto \frac{{\color{blue}{\left(1 \cdot (2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}} - \frac{x}{\tan B}\]

    8. Applied unpow-prod-down29.7

      \[\leadsto \frac{\color{blue}{{1}^{\frac{-1}{2}} \cdot {\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}} - \frac{x}{\tan B}\]

    9. Applied times-frac25.0

      \[\leadsto \color{blue}{\frac{{1}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}{\sqrt[3]{F} \cdot \sqrt[3]{F}}} \cdot \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}}} - \frac{x}{\tan B}\]

    10. Simplified25.0

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{F}}{\sqrt[3]{\sin B}} \cdot \frac{\sqrt[3]{F}}{\sqrt[3]{\sin B}}\right)} \cdot \frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sqrt[3]{\sin B}}{\sqrt[3]{F}}} - \frac{x}{\tan B}\]

    11. Taylor expanded around inf 0.1

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

    12. Simplified0.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.9108912601642475 \cdot 10^{+100}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 3.7275517711903683 \cdot 10^{+65}:\\ \;\;\;\;\frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{F \cdot F}\right) - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(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))))))