Average Error: 59.9 → 0.0
Time: 42.8s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\frac{\frac{-1}{135} \cdot \left(x \cdot x\right) + \left(\frac{1}{9} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{2025}\right)}{\frac{1}{27} + \left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}} + {x}^{5} \cdot \frac{2}{945}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{\frac{-1}{135} \cdot \left(x \cdot x\right) + \left(\frac{1}{9} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{2025}\right)}{\frac{1}{27} + \left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}} + {x}^{5} \cdot \frac{2}{945}
double f(double x) {
        double r3001439 = 1.0;
        double r3001440 = x;
        double r3001441 = r3001439 / r3001440;
        double r3001442 = tan(r3001440);
        double r3001443 = r3001439 / r3001442;
        double r3001444 = r3001441 - r3001443;
        return r3001444;
}
↓
double f(double x) {
        double r3001445 = x;
        double r3001446 = -0.007407407407407408;
        double r3001447 = r3001445 * r3001445;
        double r3001448 = r3001446 * r3001447;
        double r3001449 = 0.1111111111111111;
        double r3001450 = r3001447 * r3001447;
        double r3001451 = 0.0004938271604938272;
        double r3001452 = r3001450 * r3001451;
        double r3001453 = r3001449 + r3001452;
        double r3001454 = r3001448 + r3001453;
        double r3001455 = 0.037037037037037035;
        double r3001456 = 1.0973936899862826e-05;
        double r3001457 = r3001456 * r3001450;
        double r3001458 = r3001447 * r3001457;
        double r3001459 = r3001455 + r3001458;
        double r3001460 = r3001454 / r3001459;
        double r3001461 = r3001445 / r3001460;
        double r3001462 = 5.0;
        double r3001463 = pow(r3001445, r3001462);
        double r3001464 = 0.0021164021164021165;
        double r3001465 = r3001463 * r3001464;
        double r3001466 = r3001461 + r3001465;
        return r3001466;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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

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

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

  9. Applied associate-/l*0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 
(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))))