Average Error: 0.2 → 0.2
Time: 5.6s
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 r8845 = x;
        double r8846 = 1.0;
        double r8847 = B;
        double r8848 = tan(r8847);
        double r8849 = r8846 / r8848;
        double r8850 = r8845 * r8849;
        double r8851 = -r8850;
        double r8852 = sin(r8847);
        double r8853 = r8846 / r8852;
        double r8854 = r8851 + r8853;
        return r8854;
}
↓
double f(double B, double x) {
        double r8855 = 1.0;
        double r8856 = 1.0;
        double r8857 = x;
        double r8858 = B;
        double r8859 = cos(r8858);
        double r8860 = r8857 * r8859;
        double r8861 = r8856 - r8860;
        double r8862 = sin(r8858);
        double r8863 = r8861 / r8862;
        double r8864 = r8855 * r8863;
        return r8864;
}

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 div-inv0.2

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

  7. Applied associate-*l*0.2

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

  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 2020036 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))