Average Error: 58.2 → 0.6
Time: 36.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)}{2}
double f(double x) {
        double r13383728 = x;
        double r13383729 = exp(r13383728);
        double r13383730 = -r13383728;
        double r13383731 = exp(r13383730);
        double r13383732 = r13383729 - r13383731;
        double r13383733 = 2.0;
        double r13383734 = r13383732 / r13383733;
        return r13383734;
}
↓
double f(double x) {
        double r13383735 = 0.016666666666666666;
        double r13383736 = x;
        double r13383737 = 5.0;
        double r13383738 = pow(r13383736, r13383737);
        double r13383739 = 2.0;
        double r13383740 = r13383739 * r13383736;
        double r13383741 = 0.3333333333333333;
        double r13383742 = r13383736 * r13383736;
        double r13383743 = r13383741 * r13383742;
        double r13383744 = r13383743 * r13383736;
        double r13383745 = r13383740 + r13383744;
        double r13383746 = fma(r13383735, r13383738, r13383745);
        double r13383747 = r13383746 / r13383739;
        return r13383747;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  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

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

  5. Applied fma-udef0.6

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

  6. Applied distribute-rgt-in0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))