Average Error: 0.0 → 0.0
Time: 9.7s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{2}{e^{x} + \frac{1}{e^{x}}}\right)\right)\right)\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{2}{e^{x} + \frac{1}{e^{x}}}\right)\right)\right)\right)
double f(double x) {
        double r2232295 = 2.0;
        double r2232296 = x;
        double r2232297 = exp(r2232296);
        double r2232298 = -r2232296;
        double r2232299 = exp(r2232298);
        double r2232300 = r2232297 + r2232299;
        double r2232301 = r2232295 / r2232300;
        return r2232301;
}
↓
double f(double x) {
        double r2232302 = 2.0;
        double r2232303 = x;
        double r2232304 = exp(r2232303);
        double r2232305 = 1.0;
        double r2232306 = r2232305 / r2232304;
        double r2232307 = r2232304 + r2232306;
        double r2232308 = r2232302 / r2232307;
        double r2232309 = log1p(r2232308);
        double r2232310 = expm1(r2232309);
        return r2232310;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{2}{e^{x} + e^{-x}}\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{2}{e^{x} + e^{-x}}\right)\right)\right)\right)}\]

  4. Simplified0.0

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\mathsf{log1p}\left(\left(\frac{2}{e^{x} + \frac{1}{e^{x}}}\right)\right)\right)}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\frac{2}{e^{x} + \frac{1}{e^{x}}}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))