Average Error: 57.8 → 1.9
Time: 5.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}
double f(double x) {
        double r35176 = x;
        double r35177 = exp(r35176);
        double r35178 = -r35176;
        double r35179 = exp(r35178);
        double r35180 = r35177 - r35179;
        double r35181 = r35177 + r35179;
        double r35182 = r35180 / r35181;
        return r35182;
}
↓
double f(double x) {
        double r35183 = x;
        double r35184 = 0.13333333333333333;
        double r35185 = 5.0;
        double r35186 = pow(r35183, r35185);
        double r35187 = r35184 * r35186;
        double r35188 = r35183 + r35187;
        double r35189 = 0.3333333333333333;
        double r35190 = 3.0;
        double r35191 = pow(r35183, r35190);
        double r35192 = r35189 * r35191;
        double r35193 = r35188 - r35192;
        return r35193;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.8

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

  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. Final simplification1.9

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

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))