Average Error: 58.0 → 0.8
Time: 20.1s
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 r5498634 = x;
        double r5498635 = exp(r5498634);
        double r5498636 = -r5498634;
        double r5498637 = exp(r5498636);
        double r5498638 = r5498635 - r5498637;
        double r5498639 = 2.0;
        double r5498640 = r5498638 / r5498639;
        return r5498640;
}
↓
double f(double x) {
        double r5498641 = 0.016666666666666666;
        double r5498642 = x;
        double r5498643 = 5.0;
        double r5498644 = pow(r5498642, r5498643);
        double r5498645 = 2.0;
        double r5498646 = r5498645 * r5498642;
        double r5498647 = 0.3333333333333333;
        double r5498648 = r5498642 * r5498642;
        double r5498649 = r5498647 * r5498648;
        double r5498650 = r5498649 * r5498642;
        double r5498651 = r5498646 + r5498650;
        double r5498652 = fma(r5498641, r5498644, r5498651);
        double r5498653 = r5498652 / r5498645;
        return r5498653;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied fma-udef0.8

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

  6. Applied distribute-rgt-in0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))