Average Error: 58.0 → 0.7
Time: 4.3s
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 r55524 = x;
        double r55525 = exp(r55524);
        double r55526 = -r55524;
        double r55527 = exp(r55526);
        double r55528 = r55525 - r55527;
        double r55529 = 2.0;
        double r55530 = r55528 / r55529;
        return r55530;
}
↓
double f(double x) {
        double r55531 = 0.3333333333333333;
        double r55532 = x;
        double r55533 = 3.0;
        double r55534 = pow(r55532, r55533);
        double r55535 = 0.016666666666666666;
        double r55536 = 5.0;
        double r55537 = pow(r55532, r55536);
        double r55538 = 2.0;
        double r55539 = r55538 * r55532;
        double r55540 = fma(r55535, r55537, r55539);
        double r55541 = fma(r55531, r55534, r55540);
        double r55542 = 2.0;
        double r55543 = r55541 / r55542;
        return r55543;
}

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 2019354 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))