Average Error: 57.9 → 0.7
Time: 17.2s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}
double f(double x) {
        double r2093089 = x;
        double r2093090 = exp(r2093089);
        double r2093091 = -r2093089;
        double r2093092 = exp(r2093091);
        double r2093093 = r2093090 - r2093092;
        double r2093094 = 2.0;
        double r2093095 = r2093093 / r2093094;
        return r2093095;
}
↓
double f(double x) {
        double r2093096 = x;
        double r2093097 = 5.0;
        double r2093098 = pow(r2093096, r2093097);
        double r2093099 = 0.016666666666666666;
        double r2093100 = 0.3333333333333333;
        double r2093101 = r2093100 * r2093096;
        double r2093102 = 2.0;
        double r2093103 = fma(r2093101, r2093096, r2093102);
        double r2093104 = r2093103 * r2093096;
        double r2093105 = fma(r2093098, r2093099, r2093104);
        double r2093106 = r2093105 / r2093102;
        return r2093106;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))