Hyperbolic tangent

Average Error: 58.2 → 0.0

Time: 13.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = -2.5562774200050244e+255

Math

\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]

↓

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\tanh x\right)\right)\]

C

double f(double x) {
        double r37178 = x;
        double r37179 = exp(r37178);
        double r37180 = -r37178;
        double r37181 = exp(r37180);
        double r37182 = r37179 - r37181;
        double r37183 = r37179 + r37181;
        double r37184 = r37182 / r37183;
        return r37184;
}

↓

double f(double x) {
        double r37185 = x;
        double r37186 = tanh(r37185);
        double r37187 = log1p(r37186);
        double r37188 = expm1(r37187);
        return r37188;
}

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-undef 0.0

   \[\leadsto \color{blue}{\tanh x}\]
4. Using strategy rm

5. Applied expm1-log1p-u 0.0

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tanh x\right)\right)}\]

6. Final simplification 0.0

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\tanh x\right)\right)\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))