\[\frac{1}{x + 1} - \frac{1}{x}\]

↓

\[\frac{-1}{(x \cdot x + x)_*}\]

double f(double x) {
        double r4358344 = 1.0;
        double r4358345 = x;
        double r4358346 = r4358345 + r4358344;
        double r4358347 = r4358344 / r4358346;
        double r4358348 = r4358344 / r4358345;
        double r4358349 = r4358347 - r4358348;
        return r4358349;
}

↓

double f(double x) {
        double r4358350 = -1.0;
        double r4358351 = x;
        double r4358352 = fma(r4358351, r4358351, r4358351);
        double r4358353 = r4358350 / r4358352;
        return r4358353;
}

\frac{1}{x + 1} - \frac{1}{x}

↓

\frac{-1}{(x \cdot x + x)_*}

Derivation

Initial program 14.7
\[\frac{1}{x + 1} - \frac{1}{x}\]
Using strategy rm
Applied frac-sub14.1
\[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
Simplified0.3
\[\leadsto \frac{\color{blue}{-1}}{\left(x + 1\right) \cdot x}\]
Simplified0.3
\[\leadsto \frac{-1}{\color{blue}{(x \cdot x + x)_*}}\]
Final simplification0.3
\[\leadsto \frac{-1}{(x \cdot x + x)_*}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Error

Derivation

Reproduce