Average Error: 13.9 → 0.3
Time: 10.6s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -5.7648000314577582 \cdot 10^{21}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{\frac{1}{\sin B}}{{F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 21.3414842217350404:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{{F}^{2}}\right)\\ \end{array}\]
\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 -5.7648000314577582 \cdot 10^{21}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{\frac{1}{\sin B}}{{F}^{2}} - \frac{1}{\sin B}\right)\\

\mathbf{elif}\;F \le 21.3414842217350404:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\

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

\end{array}
double f(double F, double B, double x) {
        double r39156 = x;
        double r39157 = 1.0;
        double r39158 = B;
        double r39159 = tan(r39158);
        double r39160 = r39157 / r39159;
        double r39161 = r39156 * r39160;
        double r39162 = -r39161;
        double r39163 = F;
        double r39164 = sin(r39158);
        double r39165 = r39163 / r39164;
        double r39166 = r39163 * r39163;
        double r39167 = 2.0;
        double r39168 = r39166 + r39167;
        double r39169 = r39167 * r39156;
        double r39170 = r39168 + r39169;
        double r39171 = r39157 / r39167;
        double r39172 = -r39171;
        double r39173 = pow(r39170, r39172);
        double r39174 = r39165 * r39173;
        double r39175 = r39162 + r39174;
        return r39175;
}
↓
double f(double F, double B, double x) {
        double r39176 = F;
        double r39177 = -5.764800031457758e+21;
        bool r39178 = r39176 <= r39177;
        double r39179 = x;
        double r39180 = 1.0;
        double r39181 = r39179 * r39180;
        double r39182 = B;
        double r39183 = tan(r39182);
        double r39184 = r39181 / r39183;
        double r39185 = -r39184;
        double r39186 = sin(r39182);
        double r39187 = r39180 / r39186;
        double r39188 = 2.0;
        double r39189 = pow(r39176, r39188);
        double r39190 = r39187 / r39189;
        double r39191 = 1.0;
        double r39192 = r39191 / r39186;
        double r39193 = r39190 - r39192;
        double r39194 = r39185 + r39193;
        double r39195 = 21.34148422173504;
        bool r39196 = r39176 <= r39195;
        double r39197 = cos(r39182);
        double r39198 = r39179 * r39197;
        double r39199 = r39198 / r39186;
        double r39200 = r39180 * r39199;
        double r39201 = -r39200;
        double r39202 = r39176 / r39186;
        double r39203 = r39176 * r39176;
        double r39204 = 2.0;
        double r39205 = r39203 + r39204;
        double r39206 = r39204 * r39179;
        double r39207 = r39205 + r39206;
        double r39208 = r39180 / r39204;
        double r39209 = -r39208;
        double r39210 = pow(r39207, r39209);
        double r39211 = r39202 * r39210;
        double r39212 = r39201 + r39211;
        double r39213 = r39192 - r39190;
        double r39214 = r39185 + r39213;
        double r39215 = r39196 ? r39212 : r39214;
        double r39216 = r39178 ? r39194 : r39215;
        return r39216;
}

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 < -5.764800031457758e+21

    1. Initial program 26.4

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

    3. Applied pow-neg26.4

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

    4. Applied frac-times20.8

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

    5. Simplified20.8

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

    7. Applied associate-*r/20.7

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

    9. Applied *-un-lft-identity20.7

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

    10. Applied times-frac20.7

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

    11. Taylor expanded around -inf 0.2

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

    12. Simplified0.2

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

    if -5.764800031457758e+21 < F < 21.34148422173504

    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. Taylor expanded around inf 0.4

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

    if 21.34148422173504 < F

    1. Initial program 24.6

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

    3. Applied pow-neg24.6

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

    4. Applied frac-times18.8

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

    5. Simplified18.8

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

    7. Applied associate-*r/18.8

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

    9. Applied *-un-lft-identity18.8

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

    10. Applied times-frac18.7

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

    11. Taylor expanded around inf 0.3

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

    12. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -5.7648000314577582 \cdot 10^{21}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{\frac{1}{\sin B}}{{F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 21.3414842217350404:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{\frac{1}{\sin B}}{{F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))))