Average Error: 59.9 → 0.3
Time: 30.2s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{1}{27}}{\frac{\left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right) + \frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{1}{27}}{\frac{\left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right) + \frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}}
double f(double x) {
        double r1171392 = 1.0;
        double r1171393 = x;
        double r1171394 = r1171392 / r1171393;
        double r1171395 = tan(r1171393);
        double r1171396 = r1171392 / r1171395;
        double r1171397 = r1171394 - r1171396;
        return r1171397;
}
↓
double f(double x) {
        double r1171398 = x;
        double r1171399 = 5.0;
        double r1171400 = pow(r1171398, r1171399);
        double r1171401 = 0.0021164021164021165;
        double r1171402 = r1171400 * r1171401;
        double r1171403 = 1.0973936899862826e-05;
        double r1171404 = r1171398 * r1171398;
        double r1171405 = r1171404 * r1171398;
        double r1171406 = r1171405 * r1171405;
        double r1171407 = r1171403 * r1171406;
        double r1171408 = 0.037037037037037035;
        double r1171409 = r1171407 + r1171408;
        double r1171410 = 0.1111111111111111;
        double r1171411 = 0.007407407407407408;
        double r1171412 = r1171404 * r1171411;
        double r1171413 = r1171410 - r1171412;
        double r1171414 = 0.0004938271604938272;
        double r1171415 = r1171404 * r1171404;
        double r1171416 = r1171414 * r1171415;
        double r1171417 = r1171413 + r1171416;
        double r1171418 = r1171417 / r1171398;
        double r1171419 = r1171409 / r1171418;
        double r1171420 = r1171402 + r1171419;
        return r1171420;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 59.9

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

  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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*r/1.1

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

  7. Simplified0.2

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

  9. Applied associate-/l*0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \color{blue}{\frac{\frac{1}{27} + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{91125}}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) + \left(\frac{1}{3} \cdot \frac{1}{3} - \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \frac{1}{3}\right)}{x}}}\]

  10. Simplified0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{\frac{1}{27} + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{91125}}{\color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{2025} + \left(\frac{1}{9} - \frac{1}{135} \cdot \left(x \cdot x\right)\right)}{x}}}\]

  11. Final simplification0.3

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \frac{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{1}{27}}{\frac{\left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right) + \frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}}\]

Reproduce

herbie shell --seed 2019152 
(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))))