Average Error: 58.1 → 0.6
Time: 19.4s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right)\right)}{2}
double f(double x) {
        double r2191041 = x;
        double r2191042 = exp(r2191041);
        double r2191043 = -r2191041;
        double r2191044 = exp(r2191043);
        double r2191045 = r2191042 - r2191044;
        double r2191046 = 2.0;
        double r2191047 = r2191045 / r2191046;
        return r2191047;
}
↓
double f(double x) {
        double r2191048 = 0.016666666666666666;
        double r2191049 = x;
        double r2191050 = 5.0;
        double r2191051 = pow(r2191049, r2191050);
        double r2191052 = 0.3333333333333333;
        double r2191053 = r2191049 * r2191049;
        double r2191054 = 2.0;
        double r2191055 = fma(r2191052, r2191053, r2191054);
        double r2191056 = r2191049 * r2191055;
        double r2191057 = fma(r2191048, r2191051, r2191056);
        double r2191058 = r2191057 / r2191054;
        return r2191058;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))