Average Error: 13.7 → 0.2
Time: 33.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1.331755897157237081298683867708090760062 \cdot 10^{154}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 9.754679358530796536193448728543810818608 \cdot 10^{148}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + 1 \cdot {\left(\frac{1}{{F}^{1}}\right)}^{1}\right)} - \frac{x \cdot 1}{\tan B}\\ \end{array}\]
\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 -1.331755897157237081298683867708090760062 \cdot 10^{154}:\\
\;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 9.754679358530796536193448728543810818608 \cdot 10^{148}:\\
\;\;\;\;\frac{1}{\sin B} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r46589 = x;
        double r46590 = 1.0;
        double r46591 = B;
        double r46592 = tan(r46591);
        double r46593 = r46590 / r46592;
        double r46594 = r46589 * r46593;
        double r46595 = -r46594;
        double r46596 = F;
        double r46597 = sin(r46591);
        double r46598 = r46596 / r46597;
        double r46599 = r46596 * r46596;
        double r46600 = 2.0;
        double r46601 = r46599 + r46600;
        double r46602 = r46600 * r46589;
        double r46603 = r46601 + r46602;
        double r46604 = r46590 / r46600;
        double r46605 = -r46604;
        double r46606 = pow(r46603, r46605);
        double r46607 = r46598 * r46606;
        double r46608 = r46595 + r46607;
        return r46608;
}
↓
double f(double F, double B, double x) {
        double r46609 = F;
        double r46610 = -1.3317558971572371e+154;
        bool r46611 = r46609 <= r46610;
        double r46612 = 1.0;
        double r46613 = B;
        double r46614 = sin(r46613);
        double r46615 = 2.0;
        double r46616 = pow(r46609, r46615);
        double r46617 = r46614 * r46616;
        double r46618 = r46612 / r46617;
        double r46619 = -1.0;
        double r46620 = r46619 / r46614;
        double r46621 = r46618 + r46620;
        double r46622 = x;
        double r46623 = r46622 * r46612;
        double r46624 = tan(r46613);
        double r46625 = r46623 / r46624;
        double r46626 = r46621 - r46625;
        double r46627 = 9.754679358530797e+148;
        bool r46628 = r46609 <= r46627;
        double r46629 = 1.0;
        double r46630 = r46629 / r46614;
        double r46631 = r46609 * r46609;
        double r46632 = 2.0;
        double r46633 = r46631 + r46632;
        double r46634 = r46632 * r46622;
        double r46635 = r46633 + r46634;
        double r46636 = r46612 / r46632;
        double r46637 = pow(r46635, r46636);
        double r46638 = r46609 / r46637;
        double r46639 = r46630 * r46638;
        double r46640 = r46639 - r46625;
        double r46641 = pow(r46609, r46612);
        double r46642 = r46629 / r46641;
        double r46643 = pow(r46642, r46612);
        double r46644 = r46612 * r46643;
        double r46645 = r46609 + r46644;
        double r46646 = r46614 * r46645;
        double r46647 = r46609 / r46646;
        double r46648 = r46647 - r46625;
        double r46649 = r46628 ? r46640 : r46648;
        double r46650 = r46611 ? r46626 : r46649;
        return r46650;
}

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 < -1.3317558971572371e+154

    1. Initial program 41.3

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

    2. Simplified41.3

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

    4. Applied associate-*r/41.3

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

    5. Taylor expanded around -inf 0.2

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

    6. Simplified0.2

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

    if -1.3317558971572371e+154 < F < 9.754679358530797e+148

    1. Initial program 2.6

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

    2. Simplified2.6

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

    4. Applied associate-*r/2.5

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

    6. Applied pow-neg2.5

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

    7. Applied frac-times0.3

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

    8. Simplified0.3

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

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

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

    11. Applied times-frac0.2

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

    if 9.754679358530797e+148 < F

    1. Initial program 41.2

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

    2. Simplified41.2

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

    4. Applied associate-*r/41.1

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

    6. Applied pow-neg41.1

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

    7. Applied frac-times35.7

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

    8. Simplified35.7

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

    9. Taylor expanded around inf 0.2

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

    10. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.331755897157237081298683867708090760062 \cdot 10^{154}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 9.754679358530796536193448728543810818608 \cdot 10^{148}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + 1 \cdot {\left(\frac{1}{{F}^{1}}\right)}^{1}\right)} - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))))