Average Error: 58.1 → 0.7
Time: 4.2s
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 r64474 = x;
        double r64475 = exp(r64474);
        double r64476 = -r64474;
        double r64477 = exp(r64476);
        double r64478 = r64475 - r64477;
        double r64479 = 2.0;
        double r64480 = r64478 / r64479;
        return r64480;
}
↓
double f(double x) {
        double r64481 = 0.3333333333333333;
        double r64482 = x;
        double r64483 = 3.0;
        double r64484 = pow(r64482, r64483);
        double r64485 = 0.016666666666666666;
        double r64486 = 5.0;
        double r64487 = pow(r64482, r64486);
        double r64488 = 2.0;
        double r64489 = r64488 * r64482;
        double r64490 = fma(r64485, r64487, r64489);
        double r64491 = fma(r64481, r64484, r64490);
        double r64492 = 2.0;
        double r64493 = r64491 / r64492;
        return r64493;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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 2020024 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))