VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 5.1s

Precision: 64

Math

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

↓

\[\frac{\sqrt[3]{1} \cdot \mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\]

C

double f(double B, double x) {
        double r36625 = x;
        double r36626 = 1.0;
        double r36627 = B;
        double r36628 = tan(r36627);
        double r36629 = r36626 / r36628;
        double r36630 = r36625 * r36629;
        double r36631 = -r36630;
        double r36632 = sin(r36627);
        double r36633 = r36626 / r36632;
        double r36634 = r36631 + r36633;
        return r36634;
}

↓

double f(double B, double x) {
        double r36635 = 1.0;
        double r36636 = cbrt(r36635);
        double r36637 = B;
        double r36638 = cos(r36637);
        double r36639 = x;
        double r36640 = -r36639;
        double r36641 = 1.0;
        double r36642 = fma(r36638, r36640, r36641);
        double r36643 = r36636 * r36642;
        double r36644 = sin(r36637);
        double r36645 = r36643 / r36644;
        double r36646 = r36636 * r36636;
        double r36647 = r36645 * r36646;
        return r36647;
}

Error

Bits error versus B

Bits error versus x

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]
3. Using strategy rm

4. Applied tan-quot 0.2

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

5. Applied associate-/r/ 0.2

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

6. Taylor expanded around inf 0.2

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

7. Simplified 0.2

   \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(\cos B \cdot \left(-x\right) + 1\right)}\]
8. Using strategy rm

9. Applied *-un-lft-identity 0.2

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

10. Applied add-cube-cbrt 0.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \sin B} \cdot \left(\cos B \cdot \left(-x\right) + 1\right)\]

11. Applied times-frac 0.2

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sin B}\right)} \cdot \left(\cos B \cdot \left(-x\right) + 1\right)\]

12. Applied associate-*l* 0.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \left(\frac{\sqrt[3]{1}}{\sin B} \cdot \left(\cos B \cdot \left(-x\right) + 1\right)\right)}\]

13. Simplified 0.2

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\sqrt[3]{1} \cdot \mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}}\]

14. Final simplification 0.2

    \[\leadsto \frac{\sqrt[3]{1} \cdot \mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\]

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))))