Average Error: 0.2 → 0.1
Time: 2.4s
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 r735302 = x;
        double r735303 = r735302 * r735302;
        double r735304 = 3.0;
        double r735305 = 2.0;
        double r735306 = r735302 * r735305;
        double r735307 = r735304 - r735306;
        double r735308 = r735303 * r735307;
        return r735308;
}
↓
double f(double x) {
        double r735309 = x;
        double r735310 = r735309 * r735309;
        double r735311 = 3.0;
        double r735312 = 2.0;
        double r735313 = 3.0;
        double r735314 = pow(r735309, r735313);
        double r735315 = r735312 * r735314;
        double r735316 = -r735315;
        double r735317 = fma(r735310, r735311, r735316);
        return r735317;
}

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))))