Average Error: 0.2 → 0.2
Time: 12.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}
double f(double B, double x) {
        double r20072 = x;
        double r20073 = 1.0;
        double r20074 = B;
        double r20075 = tan(r20074);
        double r20076 = r20073 / r20075;
        double r20077 = r20072 * r20076;
        double r20078 = -r20077;
        double r20079 = sin(r20074);
        double r20080 = r20073 / r20079;
        double r20081 = r20078 + r20080;
        return r20081;
}
↓
double f(double B, double x) {
        double r20082 = 1.0;
        double r20083 = B;
        double r20084 = sin(r20083);
        double r20085 = r20082 / r20084;
        double r20086 = x;
        double r20087 = cos(r20083);
        double r20088 = r20086 * r20087;
        double r20089 = r20088 / r20084;
        double r20090 = r20082 * r20089;
        double r20091 = r20085 - r20090;
        return r20091;
}

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. Taylor expanded around inf 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))