Average Error: 0.0 → 0.8
Time: 6.1s
Precision: 64
$\frac{1}{1 + e^{-x}}$
$\frac{\sqrt{1}}{\sqrt{1 + e^{-x}}} \cdot \frac{\sqrt{1}}{\sqrt{1 + e^{-x}}}$
\frac{1}{1 + e^{-x}}
\frac{\sqrt{1}}{\sqrt{1 + e^{-x}}} \cdot \frac{\sqrt{1}}{\sqrt{1 + e^{-x}}}
double f(double x) {
double r98525 = 1.0;
double r98526 = x;
double r98527 = -r98526;
double r98528 = exp(r98527);
double r98529 = r98525 + r98528;
double r98530 = r98525 / r98529;
return r98530;
}


double f(double x) {
double r98531 = 1.0;
double r98532 = sqrt(r98531);
double r98533 = x;
double r98534 = -r98533;
double r98535 = exp(r98534);
double r98536 = r98531 + r98535;
double r98537 = sqrt(r98536);
double r98538 = r98532 / r98537;
double r98539 = r98538 * r98538;
return r98539;
}



Try it out

Results

 In Out
Enter valid numbers for all inputs

Derivation

1. Initial program 0.0

$\frac{1}{1 + e^{-x}}$
2. Using strategy rm

$\leadsto \frac{1}{\color{blue}{\sqrt{1 + e^{-x}} \cdot \sqrt{1 + e^{-x}}}}$

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

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

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

Reproduce

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