Average Error: 13.6 → 0.2
Time: 35.4s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -932124839359996544:\\ \;\;\;\;\left(-1 + \frac{1}{F \cdot F}\right) \cdot \frac{1}{\sin B} - \frac{1 \cdot x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \le 60622479.674600124359130859375:\\ \;\;\;\;{\left(F \cdot F + \left(2 \cdot x + 2\right)\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B} - \frac{1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{1 \cdot x}{\tan B}\\ \end{array}\]
\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 -932124839359996544:\\
\;\;\;\;\left(-1 + \frac{1}{F \cdot F}\right) \cdot \frac{1}{\sin B} - \frac{1 \cdot x}{\sin B} \cdot \cos B\\

\mathbf{elif}\;F \le 60622479.674600124359130859375:\\
\;\;\;\;{\left(F \cdot F + \left(2 \cdot x + 2\right)\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B} - \frac{1 \cdot x}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r61739 = x;
        double r61740 = 1.0;
        double r61741 = B;
        double r61742 = tan(r61741);
        double r61743 = r61740 / r61742;
        double r61744 = r61739 * r61743;
        double r61745 = -r61744;
        double r61746 = F;
        double r61747 = sin(r61741);
        double r61748 = r61746 / r61747;
        double r61749 = r61746 * r61746;
        double r61750 = 2.0;
        double r61751 = r61749 + r61750;
        double r61752 = r61750 * r61739;
        double r61753 = r61751 + r61752;
        double r61754 = r61740 / r61750;
        double r61755 = -r61754;
        double r61756 = pow(r61753, r61755);
        double r61757 = r61748 * r61756;
        double r61758 = r61745 + r61757;
        return r61758;
}
↓
double f(double F, double B, double x) {
        double r61759 = F;
        double r61760 = -9.321248393599965e+17;
        bool r61761 = r61759 <= r61760;
        double r61762 = -1.0;
        double r61763 = 1.0;
        double r61764 = r61759 * r61759;
        double r61765 = r61763 / r61764;
        double r61766 = r61762 + r61765;
        double r61767 = 1.0;
        double r61768 = B;
        double r61769 = sin(r61768);
        double r61770 = r61767 / r61769;
        double r61771 = r61766 * r61770;
        double r61772 = x;
        double r61773 = r61763 * r61772;
        double r61774 = r61773 / r61769;
        double r61775 = cos(r61768);
        double r61776 = r61774 * r61775;
        double r61777 = r61771 - r61776;
        double r61778 = 60622479.674600124;
        bool r61779 = r61759 <= r61778;
        double r61780 = 2.0;
        double r61781 = r61780 * r61772;
        double r61782 = r61781 + r61780;
        double r61783 = r61764 + r61782;
        double r61784 = r61763 / r61780;
        double r61785 = -r61784;
        double r61786 = pow(r61783, r61785);
        double r61787 = r61759 / r61769;
        double r61788 = r61786 * r61787;
        double r61789 = tan(r61768);
        double r61790 = r61773 / r61789;
        double r61791 = r61788 - r61790;
        double r61792 = r61765 / r61769;
        double r61793 = r61770 - r61792;
        double r61794 = r61793 - r61790;
        double r61795 = r61779 ? r61791 : r61794;
        double r61796 = r61761 ? r61777 : r61795;
        return r61796;
}

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 < -9.321248393599965e+17

    1. Initial program 25.8

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

    2. Simplified20.3

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

    4. Applied pow-neg20.2

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

    5. Applied associate-/r/20.2

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

    6. Applied *-un-lft-identity20.2

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

    7. Applied times-frac20.2

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

    8. Simplified20.2

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

    9. Simplified20.2

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

    11. Applied tan-quot20.2

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

    12. Applied associate-/r/20.2

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

    13. Simplified20.2

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

    14. Taylor expanded around -inf 0.2

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

    15. Simplified0.2

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

    if -9.321248393599965e+17 < F < 60622479.674600124

    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 div-inv0.3

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

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

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

    6. Applied times-frac0.3

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

    7. Simplified0.3

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

    9. Applied associate-*r*0.3

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

    10. Simplified0.3

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

    if 60622479.674600124 < F

    1. Initial program 24.5

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

    2. Simplified18.4

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

    4. Applied pow-neg18.4

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

    5. Applied associate-/r/18.4

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

    6. Applied *-un-lft-identity18.4

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

    7. Applied times-frac18.4

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

    8. Simplified18.4

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

    9. Simplified18.4

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

    10. Taylor expanded around inf 0.1

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

    11. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -932124839359996544:\\ \;\;\;\;\left(-1 + \frac{1}{F \cdot F}\right) \cdot \frac{1}{\sin B} - \frac{1 \cdot x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \le 60622479.674600124359130859375:\\ \;\;\;\;{\left(F \cdot F + \left(2 \cdot x + 2\right)\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B} - \frac{1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{1 \cdot x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))))