Average Error: 0.0 → 0.8
Time: 7.7s
Precision: 64
\[\frac{1}{1 + e^{-x}}\]
\[\frac{1}{\sqrt{1 + e^{-x}}} \cdot \frac{1}{\sqrt{1 + e^{-x}}}\]
\frac{1}{1 + e^{-x}}
\frac{1}{\sqrt{1 + e^{-x}}} \cdot \frac{1}{\sqrt{1 + e^{-x}}}
double f(double x) {
        double r557503 = 1.0;
        double r557504 = x;
        double r557505 = -r557504;
        double r557506 = exp(r557505);
        double r557507 = r557503 + r557506;
        double r557508 = r557503 / r557507;
        return r557508;
}

double f(double x) {
        double r557509 = 1.0;
        double r557510 = 1.0;
        double r557511 = x;
        double r557512 = -r557511;
        double r557513 = exp(r557512);
        double r557514 = r557510 + r557513;
        double r557515 = sqrt(r557514);
        double r557516 = r557509 / r557515;
        double r557517 = r557510 / r557515;
        double r557518 = r557516 * r557517;
        return r557518;
}

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

    \[\frac{1}{1 + e^{-x}}\]
  2. Using strategy rm
  3. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{1}{\color{blue}{\sqrt{1 + e^{-x}} \cdot \sqrt{1 + e^{-x}}}}\]
  4. Applied *-un-lft-identity0.8

    \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\sqrt{1 + e^{-x}} \cdot \sqrt{1 + e^{-x}}}\]
  5. Applied times-frac0.8

    \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + e^{-x}}} \cdot \frac{1}{\sqrt{1 + e^{-x}}}}\]
  6. Final simplification0.8

    \[\leadsto \frac{1}{\sqrt{1 + e^{-x}}} \cdot \frac{1}{\sqrt{1 + e^{-x}}}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "1 / (1 + exp(-x))"
  :precision binary64
  (/ 1 (+ 1 (exp (- x)))))