Average Error: 58.3 → 0.6
Time: 3.6s
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 r65526 = x;
        double r65527 = exp(r65526);
        double r65528 = -r65526;
        double r65529 = exp(r65528);
        double r65530 = r65527 - r65529;
        double r65531 = 2.0;
        double r65532 = r65530 / r65531;
        return r65532;
}
↓
double f(double x) {
        double r65533 = 0.3333333333333333;
        double r65534 = x;
        double r65535 = 3.0;
        double r65536 = pow(r65534, r65535);
        double r65537 = 0.016666666666666666;
        double r65538 = 5.0;
        double r65539 = pow(r65534, r65538);
        double r65540 = 2.0;
        double r65541 = r65540 * r65534;
        double r65542 = fma(r65537, r65539, r65541);
        double r65543 = fma(r65533, r65536, r65542);
        double r65544 = 2.0;
        double r65545 = r65543 / r65544;
        return r65545;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.3

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

  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 2020001 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))