Average Error: 13.4 → 0.3
Time: 14.4s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -2.088831848152024661481232075006803673459 \cdot 10^{45}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 29425.17876553115638671442866325378417969:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -2.088831848152024661481232075006803673459 \cdot 10^{45}:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\

\mathbf{elif}\;F \le 29425.17876553115638671442866325378417969:\\
\;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{F}{\sin B}}{{\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} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\

\end{array}
double f(double F, double B, double x) {
        double r50929 = x;
        double r50930 = 1.0;
        double r50931 = B;
        double r50932 = tan(r50931);
        double r50933 = r50930 / r50932;
        double r50934 = r50929 * r50933;
        double r50935 = -r50934;
        double r50936 = F;
        double r50937 = sin(r50931);
        double r50938 = r50936 / r50937;
        double r50939 = r50936 * r50936;
        double r50940 = 2.0;
        double r50941 = r50939 + r50940;
        double r50942 = r50940 * r50929;
        double r50943 = r50941 + r50942;
        double r50944 = r50930 / r50940;
        double r50945 = -r50944;
        double r50946 = pow(r50943, r50945);
        double r50947 = r50938 * r50946;
        double r50948 = r50935 + r50947;
        return r50948;
}
↓
double f(double F, double B, double x) {
        double r50949 = F;
        double r50950 = -2.0888318481520247e+45;
        bool r50951 = r50949 <= r50950;
        double r50952 = x;
        double r50953 = 1.0;
        double r50954 = r50952 * r50953;
        double r50955 = B;
        double r50956 = tan(r50955);
        double r50957 = r50954 / r50956;
        double r50958 = -r50957;
        double r50959 = 1.0;
        double r50960 = sin(r50955);
        double r50961 = 2.0;
        double r50962 = pow(r50949, r50961);
        double r50963 = r50960 * r50962;
        double r50964 = r50959 / r50963;
        double r50965 = r50953 * r50964;
        double r50966 = r50959 / r50960;
        double r50967 = r50965 - r50966;
        double r50968 = r50958 + r50967;
        double r50969 = 29425.178765531156;
        bool r50970 = r50949 <= r50969;
        double r50971 = r50953 / r50956;
        double r50972 = r50952 * r50971;
        double r50973 = -r50972;
        double r50974 = r50949 / r50960;
        double r50975 = r50949 * r50949;
        double r50976 = 2.0;
        double r50977 = r50975 + r50976;
        double r50978 = r50976 * r50952;
        double r50979 = r50977 + r50978;
        double r50980 = r50953 / r50976;
        double r50981 = pow(r50979, r50980);
        double r50982 = r50974 / r50981;
        double r50983 = r50973 + r50982;
        double r50984 = r50966 - r50965;
        double r50985 = r50958 + r50984;
        double r50986 = r50970 ? r50983 : r50985;
        double r50987 = r50951 ? r50968 : r50986;
        return r50987;
}

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 < -2.0888318481520247e+45

    1. Initial program 26.5

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

    3. Applied div-inv26.5

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

    4. Applied associate-*l*20.7

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

    5. Simplified20.7

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

    7. Applied associate-*r/20.7

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

    8. Taylor expanded around -inf 0.2

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

    if -2.0888318481520247e+45 < F < 29425.178765531156

    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. Using strategy rm

    3. Applied pow-neg0.5

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

    4. Applied un-div-inv0.5

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

    if 29425.178765531156 < F

    1. Initial program 25.5

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

    3. Applied div-inv25.4

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

    4. Applied associate-*l*19.4

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

    5. Simplified19.4

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

    7. Applied associate-*r/19.4

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

    8. Taylor expanded around inf 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -2.088831848152024661481232075006803673459 \cdot 10^{45}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 29425.17876553115638671442866325378417969:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(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))))))