Average Error: 0.2 → 0.1
Time: 17.0s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[3 + x \cdot \left(9 \cdot x - 12\right)\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
3 + x \cdot \left(9 \cdot x - 12\right)
double f(double x) {
        double r481489 = 3.0;
        double r481490 = x;
        double r481491 = r481490 * r481489;
        double r481492 = r481491 * r481490;
        double r481493 = 4.0;
        double r481494 = r481490 * r481493;
        double r481495 = r481492 - r481494;
        double r481496 = 1.0;
        double r481497 = r481495 + r481496;
        double r481498 = r481489 * r481497;
        return r481498;
}
↓
double f(double x) {
        double r481499 = 3.0;
        double r481500 = x;
        double r481501 = 9.0;
        double r481502 = r481501 * r481500;
        double r481503 = 12.0;
        double r481504 = r481502 - r481503;
        double r481505 = r481500 * r481504;
        double r481506 = r481499 + r481505;
        return r481506;
}

Error

Bits error versus x

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Results

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Target

Original0.2
Target0.1
Herbie0.1
\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around 0 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 
(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)))