Average Error: 0.1 → 0.1
Time: 2.2s
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 r693765 = 3.0;
        double r693766 = x;
        double r693767 = r693766 * r693765;
        double r693768 = r693767 * r693766;
        double r693769 = 4.0;
        double r693770 = r693766 * r693769;
        double r693771 = r693768 - r693770;
        double r693772 = 1.0;
        double r693773 = r693771 + r693772;
        double r693774 = r693765 * r693773;
        return r693774;
}
↓
double f(double x) {
        double r693775 = x;
        double r693776 = 9.0;
        double r693777 = r693775 * r693776;
        double r693778 = 12.0;
        double r693779 = r693777 - r693778;
        double r693780 = r693775 * r693779;
        double r693781 = 3.0;
        double r693782 = r693780 + r693781;
        return r693782;
}

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. Taylor expanded around 0 0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 
(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)))