Average Error: 59.8 → 0.3
Time: 29.2s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left({x}^{5}, \frac{2}{945}, x \cdot \frac{1}{3} + x \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left({x}^{5}, \frac{2}{945}, x \cdot \frac{1}{3} + x \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)\right)
double f(double x) {
        double r1416595 = 1.0;
        double r1416596 = x;
        double r1416597 = r1416595 / r1416596;
        double r1416598 = tan(r1416596);
        double r1416599 = r1416595 / r1416598;
        double r1416600 = r1416597 - r1416599;
        return r1416600;
}
↓
double f(double x) {
        double r1416601 = x;
        double r1416602 = 5.0;
        double r1416603 = pow(r1416601, r1416602);
        double r1416604 = 0.0021164021164021165;
        double r1416605 = 0.3333333333333333;
        double r1416606 = r1416601 * r1416605;
        double r1416607 = 0.022222222222222223;
        double r1416608 = r1416601 * r1416601;
        double r1416609 = r1416607 * r1416608;
        double r1416610 = r1416601 * r1416609;
        double r1416611 = r1416606 + r1416610;
        double r1416612 = fma(r1416603, r1416604, r1416611);
        return r1416612;
}

Error

Bits error versus x

Target

Original59.8
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 59.8

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

  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

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

  5. Applied fma-udef0.3

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

  6. Applied distribute-rgt-in0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 +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))))