Average Error: 58.0 → 0.6
Time: 11.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x\right)}{2}
double f(double x) {
        double r3408461 = x;
        double r3408462 = exp(r3408461);
        double r3408463 = -r3408461;
        double r3408464 = exp(r3408463);
        double r3408465 = r3408462 - r3408464;
        double r3408466 = 2.0;
        double r3408467 = r3408465 / r3408466;
        return r3408467;
}
↓
double f(double x) {
        double r3408468 = 0.016666666666666666;
        double r3408469 = x;
        double r3408470 = 5.0;
        double r3408471 = pow(r3408469, r3408470);
        double r3408472 = 2.0;
        double r3408473 = 0.3333333333333333;
        double r3408474 = r3408469 * r3408473;
        double r3408475 = r3408474 * r3408469;
        double r3408476 = r3408472 + r3408475;
        double r3408477 = r3408476 * r3408469;
        double r3408478 = fma(r3408468, r3408471, r3408477);
        double r3408479 = r3408478 / r3408472;
        return r3408479;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))