Average Error: 13.4 → 0.3
Time: 30.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -24245318948719908385456128:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{\frac{\frac{1}{F}}{F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 46941330.35013683140277862548828125:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -24245318948719908385456128:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{\frac{\frac{1}{F}}{F} - 1}{\sin B}\\

\mathbf{elif}\;F \le 46941330.35013683140277862548828125:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}\\

\mathbf{else}:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin 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 r50550 = x;
        double r50551 = 1.0;
        double r50552 = B;
        double r50553 = tan(r50552);
        double r50554 = r50551 / r50553;
        double r50555 = r50550 * r50554;
        double r50556 = -r50555;
        double r50557 = F;
        double r50558 = sin(r50552);
        double r50559 = r50557 / r50558;
        double r50560 = r50557 * r50557;
        double r50561 = 2.0;
        double r50562 = r50560 + r50561;
        double r50563 = r50561 * r50550;
        double r50564 = r50562 + r50563;
        double r50565 = r50551 / r50561;
        double r50566 = -r50565;
        double r50567 = pow(r50564, r50566);
        double r50568 = r50559 * r50567;
        double r50569 = r50556 + r50568;
        return r50569;
}
↓
double f(double F, double B, double x) {
        double r50570 = F;
        double r50571 = -2.424531894871991e+25;
        bool r50572 = r50570 <= r50571;
        double r50573 = 1.0;
        double r50574 = x;
        double r50575 = B;
        double r50576 = cos(r50575);
        double r50577 = r50574 * r50576;
        double r50578 = sin(r50575);
        double r50579 = r50577 / r50578;
        double r50580 = r50573 * r50579;
        double r50581 = -r50580;
        double r50582 = r50573 / r50570;
        double r50583 = r50582 / r50570;
        double r50584 = 1.0;
        double r50585 = r50583 - r50584;
        double r50586 = r50585 / r50578;
        double r50587 = r50581 + r50586;
        double r50588 = 46941330.35013683;
        bool r50589 = r50570 <= r50588;
        double r50590 = r50570 * r50570;
        double r50591 = 2.0;
        double r50592 = r50590 + r50591;
        double r50593 = r50591 * r50574;
        double r50594 = r50592 + r50593;
        double r50595 = r50573 / r50591;
        double r50596 = -r50595;
        double r50597 = pow(r50594, r50596);
        double r50598 = r50597 / r50578;
        double r50599 = r50570 * r50598;
        double r50600 = r50581 + r50599;
        double r50601 = r50584 / r50578;
        double r50602 = 2.0;
        double r50603 = pow(r50570, r50602);
        double r50604 = r50578 * r50603;
        double r50605 = r50584 / r50604;
        double r50606 = r50573 * r50605;
        double r50607 = r50601 - r50606;
        double r50608 = r50581 + r50607;
        double r50609 = r50589 ? r50600 : r50608;
        double r50610 = r50572 ? r50587 : r50609;
        return r50610;
}

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 < -2.424531894871991e+25

    1. Initial program 27.0

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

    3. Applied associate-*l/20.8

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

    4. Taylor expanded around inf 20.8

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

    5. Taylor expanded around -inf 0.2

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

    6. Simplified0.2

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

    if -2.424531894871991e+25 < F < 46941330.35013683

    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. Using strategy rm

    3. Applied associate-*l/0.4

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

    4. Taylor expanded around inf 0.3

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

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

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

    7. Applied times-frac0.3

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

    8. Simplified0.3

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

    if 46941330.35013683 < F

    1. Initial program 24.1

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

    3. Applied associate-*l/18.7

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

    4. Taylor expanded around inf 18.7

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

    5. Taylor expanded around inf 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -24245318948719908385456128:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{\frac{\frac{1}{F}}{F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 46941330.35013683140277862548828125:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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))))))