invcot (example 3.9)

Average Error: 59.9 → 0.4

Time: 18.6s

Precision: 64

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

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

Math

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

↓

\[0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(\log \left(e^{0.002116402116402116544841005563171165704262 \cdot {x}^{5}}\right) + 0.3333333333333333148296162562473909929395 \cdot x\right)\]

C

double f(double x) {
        double r95398 = 1.0;
        double r95399 = x;
        double r95400 = r95398 / r95399;
        double r95401 = tan(r95399);
        double r95402 = r95398 / r95401;
        double r95403 = r95400 - r95402;
        return r95403;
}

↓

double f(double x) {
        double r95404 = 0.022222222222222223;
        double r95405 = x;
        double r95406 = 3.0;
        double r95407 = pow(r95405, r95406);
        double r95408 = r95404 * r95407;
        double r95409 = 0.0021164021164021165;
        double r95410 = 5.0;
        double r95411 = pow(r95405, r95410);
        double r95412 = r95409 * r95411;
        double r95413 = exp(r95412);
        double r95414 = log(r95413);
        double r95415 = 0.3333333333333333;
        double r95416 = r95415 * r95405;
        double r95417 = r95414 + r95416;
        double r95418 = r95408 + r95417;
        return r95418;
}

Error

Bits error versus x

Target

Original	59.9
Target	0.1
Herbie	0.4

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

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. Using strategy rm

4. Applied add-log-exp 0.4

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

5. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019351 
(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))))