Average Error: 58.0 → 0.7
Time: 3.0s
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 r61209 = x;
        double r61210 = exp(r61209);
        double r61211 = -r61209;
        double r61212 = exp(r61211);
        double r61213 = r61210 - r61212;
        double r61214 = 2.0;
        double r61215 = r61213 / r61214;
        return r61215;
}
↓
double f(double x) {
        double r61216 = 0.3333333333333333;
        double r61217 = x;
        double r61218 = 3.0;
        double r61219 = pow(r61217, r61218);
        double r61220 = 0.016666666666666666;
        double r61221 = 5.0;
        double r61222 = pow(r61217, r61221);
        double r61223 = 2.0;
        double r61224 = r61223 * r61217;
        double r61225 = fma(r61220, r61222, r61224);
        double r61226 = fma(r61216, r61219, r61225);
        double r61227 = 2.0;
        double r61228 = r61226 / r61227;
        return r61228;
}

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 2020036 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))