Average Error: 57.9 → 0.6
Time: 6.5s
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 r29461 = x;
        double r29462 = exp(r29461);
        double r29463 = -r29461;
        double r29464 = exp(r29463);
        double r29465 = r29462 - r29464;
        double r29466 = 2.0;
        double r29467 = r29465 / r29466;
        return r29467;
}
↓
double f(double x) {
        double r29468 = 0.3333333333333333;
        double r29469 = x;
        double r29470 = 3.0;
        double r29471 = pow(r29469, r29470);
        double r29472 = 0.016666666666666666;
        double r29473 = 5.0;
        double r29474 = pow(r29469, r29473);
        double r29475 = 2.0;
        double r29476 = r29475 * r29469;
        double r29477 = fma(r29472, r29474, r29476);
        double r29478 = fma(r29468, r29471, r29477);
        double r29479 = 2.0;
        double r29480 = r29478 / r29479;
        return r29480;
}

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 2019209 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))