Average Error: 58.2 → 0.0
Time: 38.4s
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 r11690498 = x;
        double r11690499 = exp(r11690498);
        double r11690500 = -r11690498;
        double r11690501 = exp(r11690500);
        double r11690502 = r11690499 - r11690501;
        double r11690503 = r11690499 + r11690501;
        double r11690504 = r11690502 / r11690503;
        return r11690504;
}
↓
double f(double x) {
        double r11690505 = x;
        double r11690506 = tanh(r11690505);
        return r11690506;
}

Error

Bits error versus x

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Results

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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 2019128 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))