Average Error: 58.1 → 1.9
Time: 5.8s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\mathsf{fma}\left({x}^{5}, \frac{2}{15}, x - \frac{1}{3} \cdot {x}^{3}\right)\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\mathsf{fma}\left({x}^{5}, \frac{2}{15}, x - \frac{1}{3} \cdot {x}^{3}\right)
double f(double x) {
        double r38867 = x;
        double r38868 = exp(r38867);
        double r38869 = -r38867;
        double r38870 = exp(r38869);
        double r38871 = r38868 - r38870;
        double r38872 = r38868 + r38870;
        double r38873 = r38871 / r38872;
        return r38873;
}
↓
double f(double x) {
        double r38874 = x;
        double r38875 = 5.0;
        double r38876 = pow(r38874, r38875);
        double r38877 = 0.13333333333333333;
        double r38878 = 0.3333333333333333;
        double r38879 = 3.0;
        double r38880 = pow(r38874, r38879);
        double r38881 = r38878 * r38880;
        double r38882 = r38874 - r38881;
        double r38883 = fma(r38876, r38877, r38882);
        return r38883;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Simplified0.7

    \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{fma}\left(1, 1, e^{x + x}\right)}}\]

  3. Taylor expanded around 0 1.9

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

  4. Simplified1.9

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

  5. Final simplification1.9

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))