Average Error: 0.1 → 0.1
Time: 16.1s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[3 + x \cdot \left(x \cdot 9 - 12\right)\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
3 + x \cdot \left(x \cdot 9 - 12\right)
double f(double x) {
        double r575310 = 3.0;
        double r575311 = x;
        double r575312 = r575311 * r575310;
        double r575313 = r575312 * r575311;
        double r575314 = 4.0;
        double r575315 = r575311 * r575314;
        double r575316 = r575313 - r575315;
        double r575317 = 1.0;
        double r575318 = r575316 + r575317;
        double r575319 = r575310 * r575318;
        return r575319;
}
↓
double f(double x) {
        double r575320 = 3.0;
        double r575321 = x;
        double r575322 = 9.0;
        double r575323 = r575321 * r575322;
        double r575324 = 12.0;
        double r575325 = r575323 - r575324;
        double r575326 = r575321 * r575325;
        double r575327 = r575320 + r575326;
        return r575327;
}

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. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 
(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)))