Average Error: 0.1 → 0.1
Time: 3.1s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[x \cdot \left(x \cdot 9 - 12\right) + 3\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
x \cdot \left(x \cdot 9 - 12\right) + 3
double f(double x) {
        double r627902 = 3.0;
        double r627903 = x;
        double r627904 = r627903 * r627902;
        double r627905 = r627904 * r627903;
        double r627906 = 4.0;
        double r627907 = r627903 * r627906;
        double r627908 = r627905 - r627907;
        double r627909 = 1.0;
        double r627910 = r627908 + r627909;
        double r627911 = r627902 * r627910;
        return r627911;
}
↓
double f(double x) {
        double r627912 = x;
        double r627913 = 9.0;
        double r627914 = r627912 * r627913;
        double r627915 = 12.0;
        double r627916 = r627914 - r627915;
        double r627917 = r627912 * r627916;
        double r627918 = 3.0;
        double r627919 = r627917 + r627918;
        return r627919;
}

Error

Bits error versus x

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Results

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Target

Original0.1
Target0.1
Herbie0.1
\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Derivation

  1. Initial program 0.1

    \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{3 \cdot \left(1 + x \cdot \left(x \cdot 3 - 4\right)\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]

  4. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{x \cdot \left(x \cdot 9 - 12\right) + 3}\]

  6. Final simplification0.1

    \[\leadsto x \cdot \left(x \cdot 9 - 12\right) + 3\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3 (- (* (* 9 x) x) (* 12 x)))

  (* 3 (+ (- (* (* x 3) x) (* x 4)) 1)))