Average Error: 0.2 → 0.2
Time: 4.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} \cdot \left(1 - x \cdot \cos B\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} \cdot \left(1 - x \cdot \cos B\right)
double f(double B, double x) {
        double r39052 = x;
        double r39053 = 1.0;
        double r39054 = B;
        double r39055 = tan(r39054);
        double r39056 = r39053 / r39055;
        double r39057 = r39052 * r39056;
        double r39058 = -r39057;
        double r39059 = sin(r39054);
        double r39060 = r39053 / r39059;
        double r39061 = r39058 + r39060;
        return r39061;
}
↓
double f(double B, double x) {
        double r39062 = 1.0;
        double r39063 = B;
        double r39064 = sin(r39063);
        double r39065 = r39062 / r39064;
        double r39066 = 1.0;
        double r39067 = x;
        double r39068 = cos(r39063);
        double r39069 = r39067 * r39068;
        double r39070 = r39066 - r39069;
        double r39071 = r39065 * r39070;
        return r39071;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))