Average Error: 0.2 → 0.2
Time: 22.0s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \left(\cos B \cdot x\right) \cdot 1}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - \left(\cos B \cdot x\right) \cdot 1}{\sin B}
double f(double B, double x) {
        double r839824 = x;
        double r839825 = 1.0;
        double r839826 = B;
        double r839827 = tan(r839826);
        double r839828 = r839825 / r839827;
        double r839829 = r839824 * r839828;
        double r839830 = -r839829;
        double r839831 = sin(r839826);
        double r839832 = r839825 / r839831;
        double r839833 = r839830 + r839832;
        return r839833;
}
↓
double f(double B, double x) {
        double r839834 = 1.0;
        double r839835 = B;
        double r839836 = cos(r839835);
        double r839837 = x;
        double r839838 = r839836 * r839837;
        double r839839 = r839838 * r839834;
        double r839840 = r839834 - r839839;
        double r839841 = sin(r839835);
        double r839842 = r839840 / r839841;
        return r839842;
}

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded around inf 0.2

    \[\leadsto \frac{1}{\sin B} - \color{blue}{1 \cdot \frac{x \cdot \cos B}{\sin B}}\]
  4. Using strategy rm

  5. Applied associate-*r/0.2

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

  6. Applied sub-div0.2

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

  7. Final simplification0.2

    \[\leadsto \frac{1 - \left(\cos B \cdot x\right) \cdot 1}{\sin B}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))