Average Error: 0.2 → 0.2
Time: 1.7m
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 r1835774 = x;
        double r1835775 = 1.0;
        double r1835776 = B;
        double r1835777 = tan(r1835776);
        double r1835778 = r1835775 / r1835777;
        double r1835779 = r1835774 * r1835778;
        double r1835780 = -r1835779;
        double r1835781 = sin(r1835776);
        double r1835782 = r1835775 / r1835781;
        double r1835783 = r1835780 + r1835782;
        return r1835783;
}
↓
double f(double B, double x) {
        double r1835784 = 1.0;
        double r1835785 = B;
        double r1835786 = cos(r1835785);
        double r1835787 = x;
        double r1835788 = r1835786 * r1835787;
        double r1835789 = r1835784 - r1835788;
        double r1835790 = sin(r1835785);
        double r1835791 = r1835789 / r1835790;
        return r1835791;
}

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. Simplified0.1

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

  3. Taylor expanded around inf 0.2

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

  5. Applied sub-div0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))