Average Error: 60.0 → 0.2
Time: 34.9s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\sqrt{\left(\frac{1}{9} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{3}\right) + \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)}} \cdot \frac{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) + \frac{1}{27}}{\sqrt{\left(\frac{1}{9} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{3}\right) + \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)}} + {x}^{5} \cdot \frac{2}{945}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\sqrt{\left(\frac{1}{9} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{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)}} \cdot \frac{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) + \frac{1}{27}}{\sqrt{\left(\frac{1}{9} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{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)}} + {x}^{5} \cdot \frac{2}{945}
double f(double x) {
        double r2812363 = 1.0;
        double r2812364 = x;
        double r2812365 = r2812363 / r2812364;
        double r2812366 = tan(r2812364);
        double r2812367 = r2812363 / r2812366;
        double r2812368 = r2812365 - r2812367;
        return r2812368;
}
↓
double f(double x) {
        double r2812369 = x;
        double r2812370 = 0.1111111111111111;
        double r2812371 = 0.022222222222222223;
        double r2812372 = r2812369 * r2812371;
        double r2812373 = r2812369 * r2812372;
        double r2812374 = 0.3333333333333333;
        double r2812375 = r2812373 * r2812374;
        double r2812376 = r2812370 - r2812375;
        double r2812377 = r2812373 * r2812373;
        double r2812378 = r2812376 + r2812377;
        double r2812379 = sqrt(r2812378);
        double r2812380 = r2812369 / r2812379;
        double r2812381 = r2812369 * r2812369;
        double r2812382 = 1.0973936899862826e-05;
        double r2812383 = r2812382 * r2812381;
        double r2812384 = r2812381 * r2812383;
        double r2812385 = r2812384 * r2812381;
        double r2812386 = 0.037037037037037035;
        double r2812387 = r2812385 + r2812386;
        double r2812388 = r2812387 / r2812379;
        double r2812389 = r2812380 * r2812388;
        double r2812390 = 5.0;
        double r2812391 = pow(r2812369, r2812390);
        double r2812392 = 0.0021164021164021165;
        double r2812393 = r2812391 * r2812392;
        double r2812394 = r2812389 + r2812393;
        return r2812394;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 60.0

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

  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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*l/1.1

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

  7. Simplified0.3

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

  9. Applied add-sqr-sqrt0.3

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

  10. Applied times-frac0.2

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

  11. Final simplification0.2

    \[\leadsto \frac{x}{\sqrt{\left(\frac{1}{9} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{3}\right) + \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)}} \cdot \frac{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) + \frac{1}{27}}{\sqrt{\left(\frac{1}{9} - \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \frac{1}{3}\right) + \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right)}} + {x}^{5} \cdot \frac{2}{945}\]

Reproduce

herbie shell --seed 2019137 
(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))))