VandenBroeck and Keller, Equation (24)

Average Error: 0.2 → 0.2

Time: 8.7m

Precision: 64

Math

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

↓

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

C

double f(double B, double x) {
        double r17716096 = x;
        double r17716097 = 1.0;
        double r17716098 = B;
        double r17716099 = tan(r17716098);
        double r17716100 = r17716097 / r17716099;
        double r17716101 = r17716096 * r17716100;
        double r17716102 = -r17716101;
        double r17716103 = sin(r17716098);
        double r17716104 = r17716097 / r17716103;
        double r17716105 = r17716102 + r17716104;
        return r17716105;
}

↓

double f(double B, double x) {
        double r17716106 = 1.0;
        double r17716107 = B;
        double r17716108 = sin(r17716107);
        double r17716109 = r17716106 / r17716108;
        double r17716110 = x;
        double r17716111 = r17716108 / r17716110;
        double r17716112 = cos(r17716107);
        double r17716113 = r17716111 / r17716112;
        double r17716114 = r17716106 / r17716113;
        double r17716115 = r17716109 - r17716114;
        return r17716115;
}

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

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

3. Taylor expanded around -inf 0.2

   \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}}\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.6

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

6. Applied times-frac 0.6

   \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{\cos B}{\sqrt[3]{\sin B}}}\]

7. Applied add-sqr-sqrt 31.8

   \[\leadsto \frac{1}{\color{blue}{\sqrt{\sin B} \cdot \sqrt{\sin B}}} - \frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{\cos B}{\sqrt[3]{\sin B}}\]

8. Applied add-cube-cbrt 31.8

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

9. Applied times-frac 31.8

   \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sin B}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sin B}}} - \frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{\cos B}{\sqrt[3]{\sin B}}\]

10. Applied prod-diff 31.8

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

11. Simplified 0.2

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

12. Simplified 0.2

    \[\leadsto \left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \color{blue}{0}\]
13. Using strategy rm

14. Applied clear-num 0.2

    \[\leadsto \left(\frac{1}{\sin B} - \color{blue}{\frac{1}{\frac{\frac{\sin B}{x}}{\cos B}}}\right) + 0\]

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))