Average Error: 38.8 → 0.0
Time: 5.7s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[x \cdot x + x \cdot 2\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
x \cdot x + x \cdot 2
double f(double x) {
        double r16549 = x;
        double r16550 = 1.0;
        double r16551 = r16549 + r16550;
        double r16552 = r16551 * r16551;
        double r16553 = r16552 - r16550;
        return r16553;
}
↓
double f(double x) {
        double r16554 = x;
        double r16555 = r16554 * r16554;
        double r16556 = 2.0;
        double r16557 = r16554 * r16556;
        double r16558 = r16555 + r16557;
        return r16558;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 38.8

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

  2. Simplified38.8

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

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{2 \cdot x + {x}^{2}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{x \cdot \left(x + 2\right)}\]
  5. Using strategy rm

  6. Applied distribute-lft-in0.0

    \[\leadsto \color{blue}{x \cdot x + x \cdot 2}\]

  7. Simplified0.0

    \[\leadsto x \cdot x + \color{blue}{2 \cdot x}\]

  8. Final simplification0.0

    \[\leadsto x \cdot x + x \cdot 2\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))