Average Error: 0.2 → 0.2
Time: 5.1s
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 r8100 = x;
        double r8101 = 1.0;
        double r8102 = B;
        double r8103 = tan(r8102);
        double r8104 = r8101 / r8103;
        double r8105 = r8100 * r8104;
        double r8106 = -r8105;
        double r8107 = sin(r8102);
        double r8108 = r8101 / r8107;
        double r8109 = r8106 + r8108;
        return r8109;
}
↓
double f(double B, double x) {
        double r8110 = 1.0;
        double r8111 = 1.0;
        double r8112 = x;
        double r8113 = B;
        double r8114 = cos(r8113);
        double r8115 = r8112 * r8114;
        double r8116 = r8111 - r8115;
        double r8117 = r8110 * r8116;
        double r8118 = sin(r8113);
        double r8119 = r8117 / r8118;
        return r8119;
}

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. Simplified0.3

    \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(1 - x \cdot \cos B\right)}\]
  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 2019344 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))