Average Error: 0.2 → 0.2
Time: 11.5s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - 1 \cdot \left(x \cdot \cos B\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - 1 \cdot \left(x \cdot \cos B\right)}{\sin B}
double f(double B, double x) {
        double r13556 = x;
        double r13557 = 1.0;
        double r13558 = B;
        double r13559 = tan(r13558);
        double r13560 = r13557 / r13559;
        double r13561 = r13556 * r13560;
        double r13562 = -r13561;
        double r13563 = sin(r13558);
        double r13564 = r13557 / r13563;
        double r13565 = r13562 + r13564;
        return r13565;
}
↓
double f(double B, double x) {
        double r13566 = 1.0;
        double r13567 = x;
        double r13568 = B;
        double r13569 = cos(r13568);
        double r13570 = r13567 * r13569;
        double r13571 = r13566 * r13570;
        double r13572 = r13566 - r13571;
        double r13573 = sin(r13568);
        double r13574 = r13572 / r13573;
        return r13574;
}

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 associate-*r/0.2

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

  5. Taylor expanded around inf 0.2

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

  6. Simplified0.2

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

  8. Applied associate-*r/0.2

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

  9. Applied sub-div0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))