Average Error: 0.2 → 0.2
Time: 4.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \frac{1 - x \cdot \cos B}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \frac{1 - x \cdot \cos B}{\sin B}
double f(double B, double x) {
        double r8164 = x;
        double r8165 = 1.0;
        double r8166 = B;
        double r8167 = tan(r8166);
        double r8168 = r8165 / r8167;
        double r8169 = r8164 * r8168;
        double r8170 = -r8169;
        double r8171 = sin(r8166);
        double r8172 = r8165 / r8171;
        double r8173 = r8170 + r8172;
        return r8173;
}
↓
double f(double B, double x) {
        double r8174 = 1.0;
        double r8175 = 1.0;
        double r8176 = x;
        double r8177 = B;
        double r8178 = cos(r8177);
        double r8179 = r8176 * r8178;
        double r8180 = r8175 - r8179;
        double r8181 = sin(r8177);
        double r8182 = r8180 / r8181;
        double r8183 = r8174 * r8182;
        return r8183;
}

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 div-inv0.3

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

  7. Applied associate-*l*0.3

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))