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 r799380 = x;
        double r799381 = r799380 * r799380;
        double r799382 = 3.0;
        double r799383 = 2.0;
        double r799384 = r799380 * r799383;
        double r799385 = r799382 - r799384;
        double r799386 = r799381 * r799385;
        return r799386;
}
↓
double f(double x) {
        double r799387 = x;
        double r799388 = r799387 * r799387;
        double r799389 = 3.0;
        double r799390 = 2.0;
        double r799391 = 3.0;
        double r799392 = pow(r799387, r799391);
        double r799393 = r799390 * r799392;
        double r799394 = -r799393;
        double r799395 = fma(r799388, r799389, r799394);
        return r799395;
}

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 2020100 +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))))