invcot (example 3.9)

Average Error: 59.8 → 0.3

Time: 30.8s

Precision: 64

\[-0.026 \lt x \land x \lt 0.026\]

Math

\[\frac{1}{x} - \frac{1}{\tan x}\]

↓

\[\left(\left(x \cdot \sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\frac{1}{3}} \cdot \left(\sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}}} \cdot \sqrt{\sqrt{\frac{1}{3}}}\right)\right) + {x}^{5} \cdot \frac{2}{945}\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{45}\]

C

double f(double x) {
        double r4620673 = 1.0;
        double r4620674 = x;
        double r4620675 = r4620673 / r4620674;
        double r4620676 = tan(r4620674);
        double r4620677 = r4620673 / r4620676;
        double r4620678 = r4620675 - r4620677;
        return r4620678;
}

↓

double f(double x) {
        double r4620679 = x;
        double r4620680 = 0.3333333333333333;
        double r4620681 = sqrt(r4620680);
        double r4620682 = cbrt(r4620681);
        double r4620683 = r4620682 * r4620682;
        double r4620684 = sqrt(r4620683);
        double r4620685 = r4620679 * r4620684;
        double r4620686 = sqrt(r4620682);
        double r4620687 = sqrt(r4620681);
        double r4620688 = r4620686 * r4620687;
        double r4620689 = r4620681 * r4620688;
        double r4620690 = r4620685 * r4620689;
        double r4620691 = 5.0;
        double r4620692 = pow(r4620679, r4620691);
        double r4620693 = 0.0021164021164021165;
        double r4620694 = r4620692 * r4620693;
        double r4620695 = r4620690 + r4620694;
        double r4620696 = r4620679 * r4620679;
        double r4620697 = r4620696 * r4620679;
        double r4620698 = 0.022222222222222223;
        double r4620699 = r4620697 * r4620698;
        double r4620700 = r4620695 + r4620699;
        return r4620700;
}

Error

Bits error versus x

Target

Original	59.8
Target	0.1
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

1. Initial program 59.8

   \[\frac{1}{x} - \frac{1}{\tan x}\]

2. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]

3. Simplified 0.3

   \[\leadsto \color{blue}{\frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + x \cdot \frac{1}{3}\right)}\]
4. Using strategy rm

5. Applied add-sqr-sqrt 0.3

   \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + x \cdot \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \sqrt{\frac{1}{3}}\right)}\right)\]

6. Applied associate-*r* 0.7

   \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \color{blue}{\left(x \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{\frac{1}{3}}}\right)\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.7

   \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \left(x \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{3}} \cdot \sqrt{\frac{1}{3}}}}\right) \cdot \sqrt{\frac{1}{3}}\right)\]

9. Applied sqrt-prod 0.7

   \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \left(x \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{1}{3}}} \cdot \sqrt{\sqrt{\frac{1}{3}}}\right)}\right) \cdot \sqrt{\frac{1}{3}}\right)\]

10. Applied associate-*r* 0.5

    \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \color{blue}{\left(\left(x \cdot \sqrt{\sqrt{\frac{1}{3}}}\right) \cdot \sqrt{\sqrt{\frac{1}{3}}}\right)} \cdot \sqrt{\frac{1}{3}}\right)\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.5

    \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \left(\left(x \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{3}}}\right) \cdot \sqrt{\frac{1}{3}}\right)\]

13. Applied sqrt-prod 0.5

    \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \left(\left(x \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}} \cdot \sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}}}\right)}\right) \cdot \sqrt{\sqrt{\frac{1}{3}}}\right) \cdot \sqrt{\frac{1}{3}}\right)\]

14. Applied associate-*r* 0.5

    \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \left(\color{blue}{\left(\left(x \cdot \sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}}}\right)} \cdot \sqrt{\sqrt{\frac{1}{3}}}\right) \cdot \sqrt{\frac{1}{3}}\right)\]

15. Applied associate-*l* 0.4

    \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \color{blue}{\left(\left(x \cdot \sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}}} \cdot \sqrt{\sqrt{\frac{1}{3}}}\right)\right)} \cdot \sqrt{\frac{1}{3}}\right)\]

16. Applied associate-*l* 0.3

    \[\leadsto \frac{1}{45} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(\frac{2}{945} \cdot {x}^{5} + \color{blue}{\left(x \cdot \sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}}\right) \cdot \left(\left(\sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}}} \cdot \sqrt{\sqrt{\frac{1}{3}}}\right) \cdot \sqrt{\frac{1}{3}}\right)}\right)\]

17. Final simplification 0.3

    \[\leadsto \left(\left(x \cdot \sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\frac{1}{3}} \cdot \left(\sqrt{\sqrt[3]{\sqrt{\frac{1}{3}}}} \cdot \sqrt{\sqrt{\frac{1}{3}}}\right)\right) + {x}^{5} \cdot \frac{2}{945}\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{45}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))