Average Error: 0.2 → 0.2
Time: 7.3m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - \cos B \cdot x}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - \cos B \cdot x}{\sin B}
double f(double B, double x) {
        double r9413932 = x;
        double r9413933 = 1.0;
        double r9413934 = B;
        double r9413935 = tan(r9413934);
        double r9413936 = r9413933 / r9413935;
        double r9413937 = r9413932 * r9413936;
        double r9413938 = -r9413937;
        double r9413939 = sin(r9413934);
        double r9413940 = r9413933 / r9413939;
        double r9413941 = r9413938 + r9413940;
        return r9413941;
}
↓
double f(double B, double x) {
        double r9413942 = 1.0;
        double r9413943 = B;
        double r9413944 = cos(r9413943);
        double r9413945 = x;
        double r9413946 = r9413944 * r9413945;
        double r9413947 = r9413942 - r9413946;
        double r9413948 = sin(r9413943);
        double r9413949 = r9413947 / r9413948;
        return r9413949;
}

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

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
  3. Using strategy rm

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

  7. Applied associate-*l/0.2

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

  8. Applied sub-div0.2

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

  9. Final simplification0.2

    \[\leadsto \frac{1 - \cos B \cdot x}{\sin B}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))