invcot (example 3.9)

Average Error: 59.9 → 0.0

Time: 40.0s

Precision: 64

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

Math

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

↓

\[\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\frac{\mathsf{fma}\left(\frac{1}{2025}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{3}, \frac{1}{3}, \log \left(e^{\frac{-1}{45} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{3}\right)\right)}{\mathsf{fma}\left(x \cdot x, \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{1}{27}\right)}}\right)\]

C

double f(double x) {
        double r3337072 = 1.0;
        double r3337073 = x;
        double r3337074 = r3337072 / r3337073;
        double r3337075 = tan(r3337073);
        double r3337076 = r3337072 / r3337075;
        double r3337077 = r3337074 - r3337076;
        return r3337077;
}

↓

double f(double x) {
        double r3337078 = x;
        double r3337079 = 5.0;
        double r3337080 = pow(r3337078, r3337079);
        double r3337081 = 0.0021164021164021165;
        double r3337082 = 0.0004938271604938272;
        double r3337083 = r3337078 * r3337078;
        double r3337084 = r3337083 * r3337083;
        double r3337085 = 0.3333333333333333;
        double r3337086 = -0.022222222222222223;
        double r3337087 = r3337086 * r3337083;
        double r3337088 = exp(r3337087);
        double r3337089 = log(r3337088);
        double r3337090 = r3337089 * r3337085;
        double r3337091 = fma(r3337085, r3337085, r3337090);
        double r3337092 = fma(r3337082, r3337084, r3337091);
        double r3337093 = 1.0973936899862826e-05;
        double r3337094 = r3337093 * r3337083;
        double r3337095 = r3337094 * r3337083;
        double r3337096 = 0.037037037037037035;
        double r3337097 = fma(r3337083, r3337095, r3337096);
        double r3337098 = r3337092 / r3337097;
        double r3337099 = r3337078 / r3337098;
        double r3337100 = fma(r3337080, r3337081, r3337099);
        return r3337100;
}

Error

Bits error versus x

Target

Original	59.9
Target	0.1
Herbie	0.0

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

Derivation

1. Initial program 59.9

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

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

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

   \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{\color{blue}{x \cdot \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right), \left(x \cdot x\right) \cdot \frac{1}{45}, \frac{1}{27}\right)}}{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) + \left(\frac{1}{3} \cdot \frac{1}{3} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{3}\right)}\right)\]
8. Using strategy rm

9. Applied associate-/l* 0.0

   \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{945}, \color{blue}{\frac{x}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) + \left(\frac{1}{3} \cdot \frac{1}{3} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{3}\right)}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right), \left(x \cdot x\right) \cdot \frac{1}{45}, \frac{1}{27}\right)}}}\right)\]

10. Simplified 0.0

    \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2025}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{3}, \frac{1}{3}, \left(\frac{-1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right)\right)}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right), \frac{1}{27}\right)}}}\right)\]
11. Using strategy rm

12. Applied add-log-exp 0.0

    \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\frac{\mathsf{fma}\left(\frac{1}{2025}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{3}, \frac{1}{3}, \color{blue}{\log \left(e^{\frac{-1}{45} \cdot \left(x \cdot x\right)}\right)} \cdot \frac{1}{3}\right)\right)}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right), \frac{1}{27}\right)}}\right)\]

13. Final simplification 0.0

    \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\frac{\mathsf{fma}\left(\frac{1}{2025}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{3}, \frac{1}{3}, \log \left(e^{\frac{-1}{45} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{3}\right)\right)}{\mathsf{fma}\left(x \cdot x, \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{1}{27}\right)}}\right)\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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))))