Average Error: 58.0 → 0.7
Time: 3.6s
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 r60173 = x;
        double r60174 = exp(r60173);
        double r60175 = -r60173;
        double r60176 = exp(r60175);
        double r60177 = r60174 - r60176;
        double r60178 = 2.0;
        double r60179 = r60177 / r60178;
        return r60179;
}
↓
double f(double x) {
        double r60180 = 0.3333333333333333;
        double r60181 = x;
        double r60182 = 3.0;
        double r60183 = pow(r60181, r60182);
        double r60184 = 0.016666666666666666;
        double r60185 = 5.0;
        double r60186 = pow(r60181, r60185);
        double r60187 = 2.0;
        double r60188 = r60187 * r60181;
        double r60189 = fma(r60184, r60186, r60188);
        double r60190 = fma(r60180, r60183, r60189);
        double r60191 = 2.0;
        double r60192 = r60190 / r60191;
        return r60192;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))