Average Error: 58.2 → 0.6
Time: 3.1s
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 r56921 = x;
        double r56922 = exp(r56921);
        double r56923 = -r56921;
        double r56924 = exp(r56923);
        double r56925 = r56922 - r56924;
        double r56926 = 2.0;
        double r56927 = r56925 / r56926;
        return r56927;
}
↓
double f(double x) {
        double r56928 = 0.3333333333333333;
        double r56929 = x;
        double r56930 = 3.0;
        double r56931 = pow(r56929, r56930);
        double r56932 = 0.016666666666666666;
        double r56933 = 5.0;
        double r56934 = pow(r56929, r56933);
        double r56935 = 2.0;
        double r56936 = r56935 * r56929;
        double r56937 = fma(r56932, r56934, r56936);
        double r56938 = fma(r56928, r56931, r56937);
        double r56939 = 2.0;
        double r56940 = r56938 / r56939;
        return r56940;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  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 2020002 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))