Average Error: 59.7 → 0.3
Time: 8.6s
Precision: 64
\[-0.0259999999999999988 \lt x \land x \lt 0.0259999999999999988\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)
double f(double x) {
        double r93966 = 1.0;
        double r93967 = x;
        double r93968 = r93966 / r93967;
        double r93969 = tan(r93967);
        double r93970 = r93966 / r93969;
        double r93971 = r93968 - r93970;
        return r93971;
}
↓
double f(double x) {
        double r93972 = 0.022222222222222223;
        double r93973 = x;
        double r93974 = 3.0;
        double r93975 = pow(r93973, r93974);
        double r93976 = 0.0021164021164021165;
        double r93977 = 5.0;
        double r93978 = pow(r93973, r93977);
        double r93979 = 0.3333333333333333;
        double r93980 = r93979 * r93973;
        double r93981 = fma(r93976, r93978, r93980);
        double r93982 = fma(r93972, r93975, r93981);
        return r93982;
}

Error

Bits error versus x

Target

Original59.7
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.7

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{0.0222222222222222231 \cdot {x}^{3} + \left(0.00211640211640211654 \cdot {x}^{5} + 0.333333333333333315 \cdot x\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)}\]

  4. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2020045 +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))))