Average Error: 0.2 → 0.2
Time: 23.0s
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 r419445 = x;
        double r419446 = 1.0;
        double r419447 = B;
        double r419448 = tan(r419447);
        double r419449 = r419446 / r419448;
        double r419450 = r419445 * r419449;
        double r419451 = -r419450;
        double r419452 = sin(r419447);
        double r419453 = r419446 / r419452;
        double r419454 = r419451 + r419453;
        return r419454;
}
↓
double f(double B, double x) {
        double r419455 = 1.0;
        double r419456 = B;
        double r419457 = cos(r419456);
        double r419458 = x;
        double r419459 = r419457 * r419458;
        double r419460 = r419455 - r419459;
        double r419461 = sin(r419456);
        double r419462 = r419460 / r419461;
        return r419462;
}

Error

Bits error versus B

Bits error versus x

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Results

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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 2019141 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))