invcot (example 3.9)

Average Error: 59.9 → 0.3

Time: 19.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]

Math

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

↓

\[\mathsf{fma}\left(0.3333333333333333148296162562473909929395, x, \mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)\right)\]

C

double f(double x) {
        double r120586 = 1.0;
        double r120587 = x;
        double r120588 = r120586 / r120587;
        double r120589 = tan(r120587);
        double r120590 = r120586 / r120589;
        double r120591 = r120588 - r120590;
        return r120591;
}

↓

double f(double x) {
        double r120592 = 0.3333333333333333;
        double r120593 = x;
        double r120594 = 0.022222222222222223;
        double r120595 = 3.0;
        double r120596 = pow(r120593, r120595);
        double r120597 = 0.0021164021164021165;
        double r120598 = 5.0;
        double r120599 = pow(r120593, r120598);
        double r120600 = r120597 * r120599;
        double r120601 = fma(r120594, r120596, r120600);
        double r120602 = fma(r120592, r120593, r120601);
        return r120602;
}

Error

Bits error versus x

Target

Original	59.9
Target	0.1
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

1. Initial program 59.9

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

2. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)}\]

3. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, \mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, 0.3333333333333333148296162562473909929395 \cdot x\right)\right)}\]

4. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)}\]

5. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333148296162562473909929395, x, \mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)\right)}\]

6. Final simplification 0.3

   \[\leadsto \mathsf{fma}\left(0.3333333333333333148296162562473909929395, x, \mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :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))))