Average Error: 58.0 → 0.6
Time: 2.7s
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 r57468 = x;
        double r57469 = exp(r57468);
        double r57470 = -r57468;
        double r57471 = exp(r57470);
        double r57472 = r57469 - r57471;
        double r57473 = 2.0;
        double r57474 = r57472 / r57473;
        return r57474;
}
↓
double f(double x) {
        double r57475 = 0.3333333333333333;
        double r57476 = x;
        double r57477 = 3.0;
        double r57478 = pow(r57476, r57477);
        double r57479 = 0.016666666666666666;
        double r57480 = 5.0;
        double r57481 = pow(r57476, r57480);
        double r57482 = 2.0;
        double r57483 = r57482 * r57476;
        double r57484 = fma(r57479, r57481, r57483);
        double r57485 = fma(r57475, r57478, r57484);
        double r57486 = 2.0;
        double r57487 = r57485 / r57486;
        return r57487;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.6

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

  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 2020021 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))