Average Error: 0.2 → 0.1
Time: 2.9s
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 r798858 = x;
        double r798859 = r798858 * r798858;
        double r798860 = 3.0;
        double r798861 = 2.0;
        double r798862 = r798858 * r798861;
        double r798863 = r798860 - r798862;
        double r798864 = r798859 * r798863;
        return r798864;
}
↓
double f(double x) {
        double r798865 = x;
        double r798866 = r798865 * r798865;
        double r798867 = 3.0;
        double r798868 = 2.0;
        double r798869 = 3.0;
        double r798870 = pow(r798865, r798869);
        double r798871 = r798868 * r798870;
        double r798872 = -r798871;
        double r798873 = fma(r798866, r798867, r798872);
        return r798873;
}

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 2020036 +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))))