Average Error: 58.1 → 1.9
Time: 6.1s
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 r38993 = x;
        double r38994 = exp(r38993);
        double r38995 = -r38993;
        double r38996 = exp(r38995);
        double r38997 = r38994 - r38996;
        double r38998 = r38994 + r38996;
        double r38999 = r38997 / r38998;
        return r38999;
}
↓
double f(double x) {
        double r39000 = x;
        double r39001 = 5.0;
        double r39002 = pow(r39000, r39001);
        double r39003 = 0.13333333333333333;
        double r39004 = 0.3333333333333333;
        double r39005 = 3.0;
        double r39006 = pow(r39000, r39005);
        double r39007 = r39004 * r39006;
        double r39008 = r39000 - r39007;
        double r39009 = fma(r39002, r39003, r39008);
        return r39009;
}

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)))))