Average Error: 58.2 → 1.7
Time: 24.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\mathsf{fma}\left(\frac{-1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{2}{15}, {x}^{5}, x\right)\right)\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\mathsf{fma}\left(\frac{-1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{2}{15}, {x}^{5}, x\right)\right)
double f(double x) {
        double r52286 = x;
        double r52287 = exp(r52286);
        double r52288 = -r52286;
        double r52289 = exp(r52288);
        double r52290 = r52287 - r52289;
        double r52291 = r52287 + r52289;
        double r52292 = r52290 / r52291;
        return r52292;
}
↓
double f(double x) {
        double r52293 = -0.3333333333333333;
        double r52294 = x;
        double r52295 = 3.0;
        double r52296 = pow(r52294, r52295);
        double r52297 = 0.13333333333333333;
        double r52298 = 5.0;
        double r52299 = pow(r52294, r52298);
        double r52300 = fma(r52297, r52299, r52294);
        double r52301 = fma(r52293, r52296, r52300);
        return r52301;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  2. Using strategy rm

  3. Applied tanh-undef0.0

    \[\leadsto \color{blue}{\tanh x}\]

  4. Taylor expanded around 0 1.7

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

  5. Simplified1.7

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))