Average Error: 0.2 → 0.3
Time: 9.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - \left(x \cdot \frac{1}{\sin B}\right) \cdot \cos B\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \left(x \cdot \frac{1}{\sin B}\right) \cdot \cos B
double f(double B, double x) {
        double r49025 = x;
        double r49026 = 1.0;
        double r49027 = B;
        double r49028 = tan(r49027);
        double r49029 = r49026 / r49028;
        double r49030 = r49025 * r49029;
        double r49031 = -r49030;
        double r49032 = sin(r49027);
        double r49033 = r49026 / r49032;
        double r49034 = r49031 + r49033;
        return r49034;
}
↓
double f(double B, double x) {
        double r49035 = 1.0;
        double r49036 = B;
        double r49037 = sin(r49036);
        double r49038 = r49035 / r49037;
        double r49039 = x;
        double r49040 = r49039 * r49038;
        double r49041 = cos(r49036);
        double r49042 = r49040 * r49041;
        double r49043 = r49038 - r49042;
        return r49043;
}

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

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

  4. Applied tan-quot0.2

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

  5. Applied associate-/r/0.2

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

  6. Applied associate-*r*0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))