Average Error: 0.1 → 0.1
Time: 16.4s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[\left(9 \cdot x - 12\right) \cdot x + 3\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
\left(9 \cdot x - 12\right) \cdot x + 3
double f(double x) {
        double r349540 = 3.0;
        double r349541 = x;
        double r349542 = r349541 * r349540;
        double r349543 = r349542 * r349541;
        double r349544 = 4.0;
        double r349545 = r349541 * r349544;
        double r349546 = r349543 - r349545;
        double r349547 = 1.0;
        double r349548 = r349546 + r349547;
        double r349549 = r349540 * r349548;
        return r349549;
}
↓
double f(double x) {
        double r349550 = 9.0;
        double r349551 = x;
        double r349552 = r349550 * r349551;
        double r349553 = 12.0;
        double r349554 = r349552 - r349553;
        double r349555 = r349554 * r349551;
        double r349556 = 3.0;
        double r349557 = r349555 + r349556;
        return r349557;
}

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

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)}\]
  5. Using strategy rm

  6. Applied fma-udef0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(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)))