Average Error: 0.2 → 0.2
Time: 5.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 \cdot \left(1 - x \cdot \cos B\right)}{\sin B}
double f(double B, double x) {
        double r9119 = x;
        double r9120 = 1.0;
        double r9121 = B;
        double r9122 = tan(r9121);
        double r9123 = r9120 / r9122;
        double r9124 = r9119 * r9123;
        double r9125 = -r9124;
        double r9126 = sin(r9121);
        double r9127 = r9120 / r9126;
        double r9128 = r9125 + r9127;
        return r9128;
}
↓
double f(double B, double x) {
        double r9129 = 1.0;
        double r9130 = 1.0;
        double r9131 = x;
        double r9132 = B;
        double r9133 = cos(r9132);
        double r9134 = r9131 * r9133;
        double r9135 = r9130 - r9134;
        double r9136 = r9129 * r9135;
        double r9137 = sin(r9132);
        double r9138 = r9136 / r9137;
        return r9138;
}

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. Using strategy rm

  6. Applied associate-*l/0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))