Average Error: 58.2 → 0.0
Time: 3.3m
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\tanh x\]
double f(double x) {
        double r16332043 = x;
        double r16332044 = exp(r16332043);
        double r16332045 = -r16332043;
        double r16332046 = exp(r16332045);
        double r16332047 = r16332044 - r16332046;
        double r16332048 = r16332044 + r16332046;
        double r16332049 = r16332047 / r16332048;
        return r16332049;
}
↓
double f(double x) {
        double r16332050 = x;
        double r16332051 = tanh(r16332050);
        return r16332051;
}


\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\tanh x

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. Final simplification0.0

    \[\leadsto \tanh x\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))