Average Error: 58.0 → 0.6
Time: 15.0s
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 r54737 = x;
        double r54738 = exp(r54737);
        double r54739 = -r54737;
        double r54740 = exp(r54739);
        double r54741 = r54738 - r54740;
        double r54742 = 2.0;
        double r54743 = r54741 / r54742;
        return r54743;
}
↓
double f(double x) {
        double r54744 = 0.3333333333333333;
        double r54745 = x;
        double r54746 = 3.0;
        double r54747 = pow(r54745, r54746);
        double r54748 = 0.016666666666666666;
        double r54749 = 5.0;
        double r54750 = pow(r54745, r54749);
        double r54751 = 2.0;
        double r54752 = r54751 * r54745;
        double r54753 = fma(r54748, r54750, r54752);
        double r54754 = fma(r54744, r54747, r54753);
        double r54755 = 2.0;
        double r54756 = r54754 / r54755;
        return r54756;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))