Average Error: 0.2 → 0.1
Time: 2.3s
Precision: 64
\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\]
↓
\[\mathsf{fma}\left(x \cdot x, 3, -2 \cdot {x}^{3}\right)\]
\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)
↓
\mathsf{fma}\left(x \cdot x, 3, -2 \cdot {x}^{3}\right)
double f(double x) {
        double r863512 = x;
        double r863513 = r863512 * r863512;
        double r863514 = 3.0;
        double r863515 = 2.0;
        double r863516 = r863512 * r863515;
        double r863517 = r863514 - r863516;
        double r863518 = r863513 * r863517;
        return r863518;
}
↓
double f(double x) {
        double r863519 = x;
        double r863520 = r863519 * r863519;
        double r863521 = 3.0;
        double r863522 = 2.0;
        double r863523 = 3.0;
        double r863524 = pow(r863519, r863523);
        double r863525 = r863522 * r863524;
        double r863526 = -r863525;
        double r863527 = fma(r863520, r863521, r863526);
        return r863527;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 0.2

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

  3. Applied sub-neg0.2

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

  4. Applied distribute-lft-in0.2

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

  5. Simplified0.1

    \[\leadsto \left(x \cdot x\right) \cdot 3 + \color{blue}{\left(-2 \cdot {x}^{3}\right)}\]
  6. Using strategy rm

  7. Applied fma-def0.1

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

  8. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x \cdot x, 3, -2 \cdot {x}^{3}\right)\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3 (* x 2))))

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