Average Error: 39.0 → 0.0
Time: 9.8s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\left(\left(1 - \sqrt{1}\right) + x\right) \cdot \left(\left(1 + \sqrt{1}\right) + x\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\left(\left(1 - \sqrt{1}\right) + x\right) \cdot \left(\left(1 + \sqrt{1}\right) + x\right)
double f(double x) {
        double r25255 = x;
        double r25256 = 1.0;
        double r25257 = r25255 + r25256;
        double r25258 = r25257 * r25257;
        double r25259 = r25258 - r25256;
        return r25259;
}
↓
double f(double x) {
        double r25260 = 1.0;
        double r25261 = sqrt(r25260);
        double r25262 = r25260 - r25261;
        double r25263 = x;
        double r25264 = r25262 + r25263;
        double r25265 = r25260 + r25261;
        double r25266 = r25265 + r25263;
        double r25267 = r25264 * r25266;
        return r25267;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.0

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt39.0

    \[\leadsto \left(x + 1\right) \cdot \left(x + 1\right) - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\]

  4. Applied difference-of-squares39.0

    \[\leadsto \color{blue}{\left(\left(x + 1\right) + \sqrt{1}\right) \cdot \left(\left(x + 1\right) - \sqrt{1}\right)}\]

  5. Simplified39.0

    \[\leadsto \color{blue}{\left(x + \left(\sqrt{1} + 1\right)\right)} \cdot \left(\left(x + 1\right) - \sqrt{1}\right)\]

  6. Simplified0.0

    \[\leadsto \left(x + \left(\sqrt{1} + 1\right)\right) \cdot \color{blue}{\left(\left(1 - \sqrt{1}\right) + x\right)}\]

  7. Final simplification0.0

    \[\leadsto \left(\left(1 - \sqrt{1}\right) + x\right) \cdot \left(\left(1 + \sqrt{1}\right) + x\right)\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))