Average Error: 58.2 → 1.9
Time: 20.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\mathsf{fma}\left(\frac{-1}{3}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\mathsf{fma}\left(\frac{-1}{3}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)
double f(double x) {
        double r1063781 = x;
        double r1063782 = exp(r1063781);
        double r1063783 = -r1063781;
        double r1063784 = exp(r1063783);
        double r1063785 = r1063782 - r1063784;
        double r1063786 = r1063782 + r1063784;
        double r1063787 = r1063785 / r1063786;
        return r1063787;
}
↓
double f(double x) {
        double r1063788 = -0.3333333333333333;
        double r1063789 = x;
        double r1063790 = r1063789 * r1063789;
        double r1063791 = r1063790 * r1063789;
        double r1063792 = 5.0;
        double r1063793 = pow(r1063789, r1063792);
        double r1063794 = 0.13333333333333333;
        double r1063795 = fma(r1063793, r1063794, r1063789);
        double r1063796 = fma(r1063788, r1063791, r1063795);
        return r1063796;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 1.9

    \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

  3. Simplified1.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, x \cdot \left(x \cdot x\right), \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)}\]

  4. Final simplification1.9

    \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))