Average Error: 38.9 → 0.0
Time: 22.5s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[x \cdot x + 2 \cdot x\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
x \cdot x + 2 \cdot x
double f(double x) {
        double r3129 = x;
        double r3130 = 1.0;
        double r3131 = r3129 + r3130;
        double r3132 = r3131 * r3131;
        double r3133 = r3132 - r3130;
        return r3133;
}
↓
double f(double x) {
        double r3134 = x;
        double r3135 = r3134 * r3134;
        double r3136 = 2.0;
        double r3137 = r3136 * r3134;
        double r3138 = r3135 + r3137;
        return r3138;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 38.9

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  5. Applied distribute-lft-in0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))