Average Error: 58.4 → 1.6
Time: 35.6s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)
double f(double x) {
        double r2376516 = x;
        double r2376517 = exp(r2376516);
        double r2376518 = -r2376516;
        double r2376519 = exp(r2376518);
        double r2376520 = r2376517 - r2376519;
        double r2376521 = r2376517 + r2376519;
        double r2376522 = r2376520 / r2376521;
        return r2376522;
}
↓
double f(double x) {
        double r2376523 = x;
        double r2376524 = r2376523 * r2376523;
        double r2376525 = r2376523 * r2376524;
        double r2376526 = -0.3333333333333333;
        double r2376527 = 5.0;
        double r2376528 = pow(r2376523, r2376527);
        double r2376529 = 0.13333333333333333;
        double r2376530 = fma(r2376528, r2376529, r2376523);
        double r2376531 = fma(r2376525, r2376526, r2376530);
        return r2376531;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 1.6

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

  3. Simplified1.6

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

  4. Final simplification1.6

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))