Average Error: 58.0 → 0.7
Time: 39.0s
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 r79540 = x;
        double r79541 = exp(r79540);
        double r79542 = -r79540;
        double r79543 = exp(r79542);
        double r79544 = r79541 - r79543;
        double r79545 = 2.0;
        double r79546 = r79544 / r79545;
        return r79546;
}
↓
double f(double x) {
        double r79547 = 0.3333333333333333;
        double r79548 = x;
        double r79549 = 3.0;
        double r79550 = pow(r79548, r79549);
        double r79551 = 0.016666666666666666;
        double r79552 = 5.0;
        double r79553 = pow(r79548, r79552);
        double r79554 = 2.0;
        double r79555 = r79554 * r79548;
        double r79556 = fma(r79551, r79553, r79555);
        double r79557 = fma(r79547, r79550, r79556);
        double r79558 = 2.0;
        double r79559 = r79557 / r79558;
        return r79559;
}

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))