invcot (example 3.9)

Average Error: 59.9 → 0.0

Time: 29.9s

Precision: 64

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

Math

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

↓

\[\mathsf{fma}\left(\left({x}^{5}\right), \frac{2}{945}, \left(\frac{x}{\frac{\left(\frac{1}{9} - \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \frac{1}{3}\right) + \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right)}{\mathsf{fma}\left(\left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right)\right), \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right), \frac{1}{27}\right)}}\right)\right)\]

C

double f(double x) {
        double r1953013 = 1.0;
        double r1953014 = x;
        double r1953015 = r1953013 / r1953014;
        double r1953016 = tan(r1953014);
        double r1953017 = r1953013 / r1953016;
        double r1953018 = r1953015 - r1953017;
        return r1953018;
}

↓

double f(double x) {
        double r1953019 = x;
        double r1953020 = 5.0;
        double r1953021 = pow(r1953019, r1953020);
        double r1953022 = 0.0021164021164021165;
        double r1953023 = 0.1111111111111111;
        double r1953024 = 0.022222222222222223;
        double r1953025 = r1953024 * r1953019;
        double r1953026 = r1953025 * r1953019;
        double r1953027 = 0.3333333333333333;
        double r1953028 = r1953026 * r1953027;
        double r1953029 = r1953023 - r1953028;
        double r1953030 = r1953026 * r1953026;
        double r1953031 = r1953029 + r1953030;
        double r1953032 = 0.037037037037037035;
        double r1953033 = fma(r1953030, r1953026, r1953032);
        double r1953034 = r1953031 / r1953033;
        double r1953035 = r1953019 / r1953034;
        double r1953036 = fma(r1953021, r1953022, r1953035);
        return r1953036;
}

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

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied associate-/l* 0.0

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

10. Final simplification 0.0

    \[\leadsto \mathsf{fma}\left(\left({x}^{5}\right), \frac{2}{945}, \left(\frac{x}{\frac{\left(\frac{1}{9} - \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \frac{1}{3}\right) + \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right)}{\mathsf{fma}\left(\left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right)\right), \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right), \frac{1}{27}\right)}}\right)\right)\]

Reproduce

herbie shell --seed 2019132 +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))))