Average Error: 60.0 → 0.0
Time: 31.0s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(\frac{2}{945}, {x}^{5}, \frac{x}{\frac{\frac{\frac{1}{3} - \left(\frac{1}{45} \cdot x\right) \cdot x}{\mathsf{fma}\left(\frac{1}{45} \cdot x, x, \frac{1}{3}\right)}}{\frac{1}{3} - \left(\frac{1}{45} \cdot x\right) \cdot x}}\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(\frac{2}{945}, {x}^{5}, \frac{x}{\frac{\frac{\frac{1}{3} - \left(\frac{1}{45} \cdot x\right) \cdot x}{\mathsf{fma}\left(\frac{1}{45} \cdot x, x, \frac{1}{3}\right)}}{\frac{1}{3} - \left(\frac{1}{45} \cdot x\right) \cdot x}}\right)
double f(double x) {
        double r5372401 = 1.0;
        double r5372402 = x;
        double r5372403 = r5372401 / r5372402;
        double r5372404 = tan(r5372402);
        double r5372405 = r5372401 / r5372404;
        double r5372406 = r5372403 - r5372405;
        return r5372406;
}
↓
double f(double x) {
        double r5372407 = 0.0021164021164021165;
        double r5372408 = x;
        double r5372409 = 5.0;
        double r5372410 = pow(r5372408, r5372409);
        double r5372411 = 0.3333333333333333;
        double r5372412 = 0.022222222222222223;
        double r5372413 = r5372412 * r5372408;
        double r5372414 = r5372413 * r5372408;
        double r5372415 = r5372411 - r5372414;
        double r5372416 = fma(r5372413, r5372408, r5372411);
        double r5372417 = r5372415 / r5372416;
        double r5372418 = r5372417 / r5372415;
        double r5372419 = r5372408 / r5372418;
        double r5372420 = fma(r5372407, r5372410, r5372419);
        return r5372420;
}

Error

Bits error versus x

Target

Original60.0
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 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

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

  4. Taylor expanded around -inf 0.3

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

  5. Simplified0.3

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

  7. Applied flip-+0.3

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

  8. Applied associate-*r/0.3

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

  10. Applied associate-/l*0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(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))))