Average Error: 59.9 → 0.2
Time: 29.1s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{1}{27}}{\sqrt{\left(\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right) + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}} \cdot \frac{x}{\sqrt{\left(\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right) + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}}\]
\frac{1}{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}}{\sqrt{\left(\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right) + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}} \cdot \frac{x}{\sqrt{\left(\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right) + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}}
double f(double x) {
        double r1646215 = 1.0;
        double r1646216 = x;
        double r1646217 = r1646215 / r1646216;
        double r1646218 = tan(r1646216);
        double r1646219 = r1646215 / r1646218;
        double r1646220 = r1646217 - r1646219;
        return r1646220;
}
↓
double f(double x) {
        double r1646221 = x;
        double r1646222 = 5.0;
        double r1646223 = pow(r1646221, r1646222);
        double r1646224 = 0.0021164021164021165;
        double r1646225 = r1646223 * r1646224;
        double r1646226 = 1.0973936899862826e-05;
        double r1646227 = r1646221 * r1646221;
        double r1646228 = r1646227 * r1646221;
        double r1646229 = r1646228 * r1646228;
        double r1646230 = r1646226 * r1646229;
        double r1646231 = 0.037037037037037035;
        double r1646232 = r1646230 + r1646231;
        double r1646233 = 0.1111111111111111;
        double r1646234 = 0.022222222222222223;
        double r1646235 = r1646234 * r1646227;
        double r1646236 = 0.3333333333333333;
        double r1646237 = r1646235 * r1646236;
        double r1646238 = r1646233 - r1646237;
        double r1646239 = r1646235 * r1646235;
        double r1646240 = r1646238 + r1646239;
        double r1646241 = sqrt(r1646240);
        double r1646242 = r1646232 / r1646241;
        double r1646243 = r1646221 / r1646241;
        double r1646244 = r1646242 * r1646243;
        double r1646245 = r1646225 + r1646244;
        return r1646245;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 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. Simplified0.3

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

  9. Applied add-sqr-sqrt0.3

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

  10. Applied times-frac0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 
(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))))