Average Error: 0.3 → 0.2
Time: 9.0s
Precision: 64
\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x\]
↓
\[6 \cdot x + \left(-9\right) \cdot \left(x \cdot x\right)\]
\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x
↓
6 \cdot x + \left(-9\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r702552 = 3.0;
        double r702553 = 2.0;
        double r702554 = x;
        double r702555 = r702554 * r702552;
        double r702556 = r702553 - r702555;
        double r702557 = r702552 * r702556;
        double r702558 = r702557 * r702554;
        return r702558;
}
↓
double f(double x) {
        double r702559 = 6.0;
        double r702560 = x;
        double r702561 = r702559 * r702560;
        double r702562 = 9.0;
        double r702563 = -r702562;
        double r702564 = r702560 * r702560;
        double r702565 = r702563 * r702564;
        double r702566 = r702561 + r702565;
        return r702566;
}

Error

Bits error versus x

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Results

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Target

Original0.3
Target0.2
Herbie0.2
\[6 \cdot x - 9 \cdot \left(x \cdot x\right)\]

Derivation

  1. Initial program 0.3

    \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x\]
  2. Using strategy rm

  3. Applied associate-*l*0.3

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{6 \cdot x - 9 \cdot {x}^{2}}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{x \cdot \left(6 - x \cdot 9\right)}\]
  6. Using strategy rm

  7. Applied sub-neg0.2

    \[\leadsto x \cdot \color{blue}{\left(6 + \left(-x \cdot 9\right)\right)}\]

  8. Applied distribute-lft-in0.2

    \[\leadsto \color{blue}{x \cdot 6 + x \cdot \left(-x \cdot 9\right)}\]

  9. Simplified0.2

    \[\leadsto \color{blue}{6 \cdot x} + x \cdot \left(-x \cdot 9\right)\]

  10. Simplified0.2

    \[\leadsto 6 \cdot x + \color{blue}{\left(-9\right) \cdot \left(x \cdot x\right)}\]

  11. Final simplification0.2

    \[\leadsto 6 \cdot x + \left(-9\right) \cdot \left(x \cdot x\right)\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6 x) (* 9 (* x x)))

  (* (* 3 (- 2 (* x 3))) x))