Average Error: 13.0 → 0.2
Time: 1.5m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -455795803.3194169:\\ \;\;\;\;\frac{1}{-\left(\frac{\sin B}{F \cdot F} \cdot x + \sin B\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 51872366.5879824:\\ \;\;\;\;\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot F} \cdot x + \sin B} - \frac{x}{\tan B}\\ \end{array}\]
\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 -455795803.3194169:\\
\;\;\;\;\frac{1}{-\left(\frac{\sin B}{F \cdot F} \cdot x + \sin B\right)} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \le 51872366.5879824:\\
\;\;\;\;\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\sin B}{F \cdot F} \cdot x + \sin B} - \frac{x}{\tan B}\\

\end{array}
double f(double F, double B, double x) {
        double r5370047 = x;
        double r5370048 = 1.0;
        double r5370049 = B;
        double r5370050 = tan(r5370049);
        double r5370051 = r5370048 / r5370050;
        double r5370052 = r5370047 * r5370051;
        double r5370053 = -r5370052;
        double r5370054 = F;
        double r5370055 = sin(r5370049);
        double r5370056 = r5370054 / r5370055;
        double r5370057 = r5370054 * r5370054;
        double r5370058 = 2.0;
        double r5370059 = r5370057 + r5370058;
        double r5370060 = r5370058 * r5370047;
        double r5370061 = r5370059 + r5370060;
        double r5370062 = r5370048 / r5370058;
        double r5370063 = -r5370062;
        double r5370064 = pow(r5370061, r5370063);
        double r5370065 = r5370056 * r5370064;
        double r5370066 = r5370053 + r5370065;
        return r5370066;
}
↓
double f(double F, double B, double x) {
        double r5370067 = F;
        double r5370068 = -455795803.3194169;
        bool r5370069 = r5370067 <= r5370068;
        double r5370070 = 1.0;
        double r5370071 = B;
        double r5370072 = sin(r5370071);
        double r5370073 = r5370067 * r5370067;
        double r5370074 = r5370072 / r5370073;
        double r5370075 = x;
        double r5370076 = r5370074 * r5370075;
        double r5370077 = r5370076 + r5370072;
        double r5370078 = -r5370077;
        double r5370079 = r5370070 / r5370078;
        double r5370080 = tan(r5370071);
        double r5370081 = r5370075 / r5370080;
        double r5370082 = r5370079 - r5370081;
        double r5370083 = 51872366.5879824;
        bool r5370084 = r5370067 <= r5370083;
        double r5370085 = 2.0;
        double r5370086 = r5370085 * r5370075;
        double r5370087 = r5370073 + r5370085;
        double r5370088 = r5370086 + r5370087;
        double r5370089 = -0.5;
        double r5370090 = pow(r5370088, r5370089);
        double r5370091 = r5370072 / r5370067;
        double r5370092 = r5370090 / r5370091;
        double r5370093 = r5370092 - r5370081;
        double r5370094 = r5370070 / r5370077;
        double r5370095 = r5370094 - r5370081;
        double r5370096 = r5370084 ? r5370093 : r5370095;
        double r5370097 = r5370069 ? r5370082 : r5370096;
        return r5370097;
}

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 < -455795803.3194169

    1. Initial program 23.9

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

    2. Simplified19.6

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

    4. Applied clear-num19.6

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

    5. Taylor expanded around -inf 0.2

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

    6. Simplified0.2

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

    if -455795803.3194169 < F < 51872366.5879824

    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

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

    4. Applied associate-/l*0.3

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

    if 51872366.5879824 < 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. Simplified19.9

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

    4. Applied clear-num19.9

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -455795803.3194169:\\ \;\;\;\;\frac{1}{-\left(\frac{\sin B}{F \cdot F} \cdot x + \sin B\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 51872366.5879824:\\ \;\;\;\;\frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot F} \cdot x + \sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))