Average Error: 13.9 → 0.8
Time: 10.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -4.5035968290748607 \cdot 10^{47}:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 5.75706281603868059 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - 1 \cdot \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array}\]
\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 -4.5035968290748607 \cdot 10^{47}:\\
\;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - x \cdot \frac{1}{\tan B}\\

\mathbf{elif}\;F \le 5.75706281603868059 \cdot 10^{-13}:\\
\;\;\;\;F \cdot \frac{1}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - 1 \cdot \frac{x}{\frac{\sin B}{\cos B}}\\

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

\end{array}
double f(double F, double B, double x) {
        double r36992 = x;
        double r36993 = 1.0;
        double r36994 = B;
        double r36995 = tan(r36994);
        double r36996 = r36993 / r36995;
        double r36997 = r36992 * r36996;
        double r36998 = -r36997;
        double r36999 = F;
        double r37000 = sin(r36994);
        double r37001 = r36999 / r37000;
        double r37002 = r36999 * r36999;
        double r37003 = 2.0;
        double r37004 = r37002 + r37003;
        double r37005 = r37003 * r36992;
        double r37006 = r37004 + r37005;
        double r37007 = r36993 / r37003;
        double r37008 = -r37007;
        double r37009 = pow(r37006, r37008);
        double r37010 = r37001 * r37009;
        double r37011 = r36998 + r37010;
        return r37011;
}
↓
double f(double F, double B, double x) {
        double r37012 = F;
        double r37013 = -4.503596829074861e+47;
        bool r37014 = r37012 <= r37013;
        double r37015 = 1.0;
        double r37016 = 1.0;
        double r37017 = B;
        double r37018 = sin(r37017);
        double r37019 = 2.0;
        double r37020 = pow(r37012, r37019);
        double r37021 = r37018 * r37020;
        double r37022 = r37016 / r37021;
        double r37023 = r37015 * r37022;
        double r37024 = r37016 / r37018;
        double r37025 = r37023 - r37024;
        double r37026 = x;
        double r37027 = tan(r37017);
        double r37028 = r37015 / r37027;
        double r37029 = r37026 * r37028;
        double r37030 = r37025 - r37029;
        double r37031 = 5.757062816038681e-13;
        bool r37032 = r37012 <= r37031;
        double r37033 = r37012 * r37012;
        double r37034 = 2.0;
        double r37035 = r37033 + r37034;
        double r37036 = r37034 * r37026;
        double r37037 = r37035 + r37036;
        double r37038 = r37015 / r37034;
        double r37039 = pow(r37037, r37038);
        double r37040 = r37018 * r37039;
        double r37041 = r37016 / r37040;
        double r37042 = r37012 * r37041;
        double r37043 = cos(r37017);
        double r37044 = r37018 / r37043;
        double r37045 = r37026 / r37044;
        double r37046 = r37015 * r37045;
        double r37047 = r37042 - r37046;
        double r37048 = r37024 - r37023;
        double r37049 = r37048 - r37029;
        double r37050 = r37032 ? r37047 : r37049;
        double r37051 = r37014 ? r37030 : r37050;
        return r37051;
}

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 < -4.503596829074861e+47

    1. Initial program 29.0

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

    2. Simplified29.0

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

    3. Taylor expanded around -inf 0.2

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

    if -4.503596829074861e+47 < F < 5.757062816038681e-13

    1. Initial program 0.5

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

    2. Simplified0.5

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

    4. Applied pow-neg0.6

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

    5. Applied frac-times0.4

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

    6. Simplified0.4

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

    7. Taylor expanded around inf 0.3

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

    9. Applied associate-/l*0.3

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

    11. Applied div-inv0.3

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

    if 5.757062816038681e-13 < F

    1. Initial program 24.2

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

    2. Simplified24.2

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

    3. Taylor expanded around inf 2.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -4.5035968290748607 \cdot 10^{47}:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 5.75706281603868059 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - 1 \cdot \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))))