Average Error: 59.9 → 0.0
Time: 44.9s
Precision: 64
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\frac{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}} + \frac{2}{945} \cdot {x}^{5}\]
double f(double x) {
        double r4903341 = 1.0;
        double r4903342 = x;
        double r4903343 = r4903341 / r4903342;
        double r4903344 = tan(r4903342);
        double r4903345 = r4903341 / r4903344;
        double r4903346 = r4903343 - r4903345;
        return r4903346;
}
↓
double f(double x) {
        double r4903347 = x;
        double r4903348 = 0.3333333333333333;
        double r4903349 = 0.022222222222222223;
        double r4903350 = r4903347 * r4903347;
        double r4903351 = r4903349 * r4903350;
        double r4903352 = r4903348 - r4903351;
        double r4903353 = 0.1111111111111111;
        double r4903354 = r4903351 * r4903351;
        double r4903355 = r4903353 - r4903354;
        double r4903356 = r4903352 / r4903355;
        double r4903357 = r4903347 / r4903356;
        double r4903358 = 0.0021164021164021165;
        double r4903359 = 5.0;
        double r4903360 = pow(r4903347, r4903359);
        double r4903361 = r4903358 * r4903360;
        double r4903362 = r4903357 + r4903361;
        return r4903362;
}


\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}} + \frac{2}{945} \cdot {x}^{5}

Error

Bits error versus x

Target

Original59.9
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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. Simplified0.3

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

  5. Applied flip-+0.3

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

  6. Applied associate-*r/0.3

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

  8. Applied associate-/l*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019102 
(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))))