Average Error: 58.2 → 1.9
Time: 20.6s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\left(x + \left(x \cdot \frac{-1}{3}\right) \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{15}\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\left(x + \left(x \cdot \frac{-1}{3}\right) \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{15}
double f(double x) {
        double r1216528 = x;
        double r1216529 = exp(r1216528);
        double r1216530 = -r1216528;
        double r1216531 = exp(r1216530);
        double r1216532 = r1216529 - r1216531;
        double r1216533 = r1216529 + r1216531;
        double r1216534 = r1216532 / r1216533;
        return r1216534;
}
↓
double f(double x) {
        double r1216535 = x;
        double r1216536 = -0.3333333333333333;
        double r1216537 = r1216535 * r1216536;
        double r1216538 = r1216535 * r1216535;
        double r1216539 = r1216537 * r1216538;
        double r1216540 = r1216535 + r1216539;
        double r1216541 = 5.0;
        double r1216542 = pow(r1216535, r1216541);
        double r1216543 = 0.13333333333333333;
        double r1216544 = r1216542 * r1216543;
        double r1216545 = r1216540 + r1216544;
        return r1216545;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 1.9

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

  5. Simplified1.9

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

  6. Final simplification1.9

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))