VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 5.8s

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 r9664 = x;
        double r9665 = 1.0;
        double r9666 = B;
        double r9667 = tan(r9666);
        double r9668 = r9665 / r9667;
        double r9669 = r9664 * r9668;
        double r9670 = -r9669;
        double r9671 = sin(r9666);
        double r9672 = r9665 / r9671;
        double r9673 = r9670 + r9672;
        return r9673;
}

↓

double f(double B, double x) {
        double r9674 = 1.0;
        double r9675 = cbrt(r9674);
        double r9676 = B;
        double r9677 = cos(r9676);
        double r9678 = x;
        double r9679 = -r9678;
        double r9680 = 1.0;
        double r9681 = fma(r9677, r9679, r9680);
        double r9682 = r9675 * r9681;
        double r9683 = sin(r9676);
        double r9684 = r9682 / r9683;
        double r9685 = r9675 * r9675;
        double r9686 = r9684 * r9685;
        return r9686;
}

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 2020035 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))