Average Error: 58.3 → 0.5
Time: 25.5s
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 r1539117 = x;
        double r1539118 = exp(r1539117);
        double r1539119 = -r1539117;
        double r1539120 = exp(r1539119);
        double r1539121 = r1539118 - r1539120;
        double r1539122 = 2.0;
        double r1539123 = r1539121 / r1539122;
        return r1539123;
}
↓
double f(double x) {
        double r1539124 = x;
        double r1539125 = 5.0;
        double r1539126 = pow(r1539124, r1539125);
        double r1539127 = 0.016666666666666666;
        double r1539128 = 0.3333333333333333;
        double r1539129 = r1539128 * r1539124;
        double r1539130 = 2.0;
        double r1539131 = fma(r1539129, r1539124, r1539130);
        double r1539132 = r1539131 * r1539124;
        double r1539133 = fma(r1539126, r1539127, r1539132);
        double r1539134 = r1539133 / r1539130;
        return r1539134;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))