Average Error: 58.0 → 0.6
Time: 4.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}
double f(double x) {
        double r38292 = x;
        double r38293 = exp(r38292);
        double r38294 = -r38292;
        double r38295 = exp(r38294);
        double r38296 = r38293 - r38295;
        double r38297 = 2.0;
        double r38298 = r38296 / r38297;
        return r38298;
}
↓
double f(double x) {
        double r38299 = 0.3333333333333333;
        double r38300 = x;
        double r38301 = 3.0;
        double r38302 = pow(r38300, r38301);
        double r38303 = 0.016666666666666666;
        double r38304 = 5.0;
        double r38305 = pow(r38300, r38304);
        double r38306 = 2.0;
        double r38307 = r38306 * r38300;
        double r38308 = fma(r38303, r38305, r38307);
        double r38309 = fma(r38299, r38302, r38308);
        double r38310 = 2.0;
        double r38311 = r38309 / r38310;
        return r38311;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2}\]

  3. Simplified0.6

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}}{2}\]

  4. Final simplification0.6

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))