Average Error: 58.2 → 0.0
Time: 20.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\tanh x\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\tanh x
double f(double x) {
        double r2894370 = x;
        double r2894371 = exp(r2894370);
        double r2894372 = -r2894370;
        double r2894373 = exp(r2894372);
        double r2894374 = r2894371 - r2894373;
        double r2894375 = r2894371 + r2894373;
        double r2894376 = r2894374 / r2894375;
        return r2894376;
}
↓
double f(double x) {
        double r2894377 = x;
        double r2894378 = tanh(r2894377);
        return r2894378;
}

Error

Bits error versus x

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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 inf 58.2

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}}\]

  5. Simplified0.0

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

  6. Final simplification0.0

    \[\leadsto \tanh x\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))