Average Error: 58.1 → 0.0
Time: 20.4s
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 r1174220 = x;
        double r1174221 = exp(r1174220);
        double r1174222 = -r1174220;
        double r1174223 = exp(r1174222);
        double r1174224 = r1174221 - r1174223;
        double r1174225 = r1174221 + r1174223;
        double r1174226 = r1174224 / r1174225;
        return r1174226;
}
↓
double f(double x) {
        double r1174227 = x;
        double r1174228 = tanh(r1174227);
        return r1174228;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 58.1

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
  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 2019141 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))