Average Error: 0.2 → 0.2
Time: 21.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \left(\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\right)\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \left(\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\right)
double f(double B, double x) {
        double r23042 = x;
        double r23043 = 1.0;
        double r23044 = B;
        double r23045 = tan(r23044);
        double r23046 = r23043 / r23045;
        double r23047 = r23042 * r23046;
        double r23048 = -r23047;
        double r23049 = sin(r23044);
        double r23050 = r23043 / r23049;
        double r23051 = r23048 + r23050;
        return r23051;
}
↓
double f(double B, double x) {
        double r23052 = 1.0;
        double r23053 = 1.0;
        double r23054 = B;
        double r23055 = sin(r23054);
        double r23056 = r23053 / r23055;
        double r23057 = x;
        double r23058 = cos(r23054);
        double r23059 = r23057 * r23058;
        double r23060 = r23059 / r23055;
        double r23061 = r23056 - r23060;
        double r23062 = r23052 * r23061;
        return r23062;
}

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. Taylor expanded around inf 0.2

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

  3. Simplified0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))