Average Error: 58.0 → 0.0
Time: 16.6s
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 r1048672 = x;
        double r1048673 = exp(r1048672);
        double r1048674 = -r1048672;
        double r1048675 = exp(r1048674);
        double r1048676 = r1048673 - r1048675;
        double r1048677 = r1048673 + r1048675;
        double r1048678 = r1048676 / r1048677;
        return r1048678;
}
↓
double f(double x) {
        double r1048679 = x;
        double r1048680 = tanh(r1048679);
        return r1048680;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 58.0

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2019154 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))