Average Error: 57.9 → 0.6
Time: 39.7s
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 r39367 = x;
        double r39368 = exp(r39367);
        double r39369 = -r39367;
        double r39370 = exp(r39369);
        double r39371 = r39368 - r39370;
        double r39372 = 2.0;
        double r39373 = r39371 / r39372;
        return r39373;
}
↓
double f(double x) {
        double r39374 = 0.3333333333333333;
        double r39375 = x;
        double r39376 = 3.0;
        double r39377 = pow(r39375, r39376);
        double r39378 = 0.016666666666666666;
        double r39379 = 5.0;
        double r39380 = pow(r39375, r39379);
        double r39381 = 2.0;
        double r39382 = r39381 * r39375;
        double r39383 = fma(r39378, r39380, r39382);
        double r39384 = fma(r39374, r39377, r39383);
        double r39385 = 2.0;
        double r39386 = r39384 / r39385;
        return r39386;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))