Average Error: 58.1 → 0.7
Time: 7.7s
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 r34666 = x;
        double r34667 = exp(r34666);
        double r34668 = -r34666;
        double r34669 = exp(r34668);
        double r34670 = r34667 - r34669;
        double r34671 = 2.0;
        double r34672 = r34670 / r34671;
        return r34672;
}
↓
double f(double x) {
        double r34673 = 0.3333333333333333;
        double r34674 = x;
        double r34675 = 3.0;
        double r34676 = pow(r34674, r34675);
        double r34677 = 0.016666666666666666;
        double r34678 = 5.0;
        double r34679 = pow(r34674, r34678);
        double r34680 = 2.0;
        double r34681 = r34680 * r34674;
        double r34682 = fma(r34677, r34679, r34681);
        double r34683 = fma(r34673, r34676, r34682);
        double r34684 = 2.0;
        double r34685 = r34683 / r34684;
        return r34685;
}

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 2019305 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))