Average Error: 57.8 → 0.7
Time: 16.8s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right), x, {x}^{5} \cdot \frac{1}{60}\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right), x, {x}^{5} \cdot \frac{1}{60}\right)}{2}
double f(double x) {
        double r3001201 = x;
        double r3001202 = exp(r3001201);
        double r3001203 = -r3001201;
        double r3001204 = exp(r3001203);
        double r3001205 = r3001202 - r3001204;
        double r3001206 = 2.0;
        double r3001207 = r3001205 / r3001206;
        return r3001207;
}
↓
double f(double x) {
        double r3001208 = 0.3333333333333333;
        double r3001209 = x;
        double r3001210 = r3001209 * r3001209;
        double r3001211 = 2.0;
        double r3001212 = fma(r3001208, r3001210, r3001211);
        double r3001213 = 5.0;
        double r3001214 = pow(r3001209, r3001213);
        double r3001215 = 0.016666666666666666;
        double r3001216 = r3001214 * r3001215;
        double r3001217 = fma(r3001212, r3001209, r3001216);
        double r3001218 = r3001217 / r3001211;
        return r3001218;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.8

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

  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

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

  4. Taylor expanded around 0 0.7

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

  5. Simplified0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))