Average Error: 14.2 → 0.3
Time: 19.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1.5313961451328438 \cdot 10^{69}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \le 1.05767038865912844 \cdot 10^{27}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{F}{\sin B}}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -1.5313961451328438 \cdot 10^{69}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right)\\

\mathbf{elif}\;F \le 1.05767038865912844 \cdot 10^{27}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{F}{\sin B}}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}}\\

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

\end{array}
double f(double F, double B, double x) {
        double r65004 = x;
        double r65005 = 1.0;
        double r65006 = B;
        double r65007 = tan(r65006);
        double r65008 = r65005 / r65007;
        double r65009 = r65004 * r65008;
        double r65010 = -r65009;
        double r65011 = F;
        double r65012 = sin(r65006);
        double r65013 = r65011 / r65012;
        double r65014 = r65011 * r65011;
        double r65015 = 2.0;
        double r65016 = r65014 + r65015;
        double r65017 = r65015 * r65004;
        double r65018 = r65016 + r65017;
        double r65019 = r65005 / r65015;
        double r65020 = -r65019;
        double r65021 = pow(r65018, r65020);
        double r65022 = r65013 * r65021;
        double r65023 = r65010 + r65022;
        return r65023;
}
↓
double f(double F, double B, double x) {
        double r65024 = F;
        double r65025 = -1.5313961451328438e+69;
        bool r65026 = r65024 <= r65025;
        double r65027 = x;
        double r65028 = 1.0;
        double r65029 = r65027 * r65028;
        double r65030 = B;
        double r65031 = tan(r65030);
        double r65032 = r65029 / r65031;
        double r65033 = -r65032;
        double r65034 = sin(r65030);
        double r65035 = 2.0;
        double r65036 = pow(r65024, r65035);
        double r65037 = r65034 * r65036;
        double r65038 = r65028 / r65037;
        double r65039 = -1.0;
        double r65040 = r65039 / r65034;
        double r65041 = r65038 + r65040;
        double r65042 = r65033 + r65041;
        double r65043 = 1.0576703886591284e+27;
        bool r65044 = r65024 <= r65043;
        double r65045 = r65024 / r65034;
        double r65046 = 2.0;
        double r65047 = fma(r65024, r65024, r65046);
        double r65048 = fma(r65046, r65027, r65047);
        double r65049 = r65028 / r65046;
        double r65050 = pow(r65048, r65049);
        double r65051 = r65045 / r65050;
        double r65052 = r65033 + r65051;
        double r65053 = r65028 / r65031;
        double r65054 = r65027 * r65053;
        double r65055 = -r65054;
        double r65056 = 1.0;
        double r65057 = r65056 / r65034;
        double r65058 = r65057 - r65038;
        double r65059 = r65055 + r65058;
        double r65060 = r65044 ? r65052 : r65059;
        double r65061 = r65026 ? r65042 : r65060;
        return r65061;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -1.5313961451328438e+69

    1. Initial program 31.0

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

    3. Applied div-inv30.9

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

    4. Applied associate-*l*24.0

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

    5. Simplified24.1

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

    7. Applied associate-*r/24.0

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

    8. Taylor expanded around -inf 0.1

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

    9. Simplified0.1

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

    if -1.5313961451328438e+69 < F < 1.0576703886591284e+27

    1. Initial program 0.6

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

    3. Applied div-inv0.6

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

    4. Applied associate-*l*0.4

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

    5. Simplified0.4

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

    7. Applied associate-*r/0.3

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

    9. Applied div-inv0.3

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

    11. Applied pow-neg0.3

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

    12. Applied associate-*l/0.3

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

    13. Applied associate-*r/0.5

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

    14. Simplified0.5

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

    if 1.0576703886591284e+27 < F

    1. Initial program 27.2

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.5313961451328438 \cdot 10^{69}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \le 1.05767038865912844 \cdot 10^{27}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{\frac{F}{\sin B}}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))))