Average Error: 0.4 → 0.2
Time: 20.9s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\left(4 \cdot y - 3 \cdot x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\left(4 \cdot y - 3 \cdot x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)
double f(double x, double y, double z) {
        double r167217 = x;
        double r167218 = y;
        double r167219 = r167218 - r167217;
        double r167220 = 6.0;
        double r167221 = r167219 * r167220;
        double r167222 = 2.0;
        double r167223 = 3.0;
        double r167224 = r167222 / r167223;
        double r167225 = z;
        double r167226 = r167224 - r167225;
        double r167227 = r167221 * r167226;
        double r167228 = r167217 + r167227;
        return r167228;
}
↓
double f(double x, double y, double z) {
        double r167229 = 4.0;
        double r167230 = y;
        double r167231 = r167229 * r167230;
        double r167232 = 3.0;
        double r167233 = x;
        double r167234 = r167232 * r167233;
        double r167235 = r167231 - r167234;
        double r167236 = r167230 - r167233;
        double r167237 = 6.0;
        double r167238 = r167236 * r167237;
        double r167239 = z;
        double r167240 = -r167239;
        double r167241 = r167238 * r167240;
        double r167242 = r167235 + r167241;
        return r167242;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
  2. Using strategy rm

  3. Applied sub-neg0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\frac{2}{3} + \left(-z\right)\right)}\]

  4. Applied distribute-lft-in0.4

    \[\leadsto x + \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\right)}\]

  5. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(x + \left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3}\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)}\]

  6. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3}, \left(y - x\right) \cdot 6, x\right)} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]

  7. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(4 \cdot y - 3 \cdot x\right)} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]

  8. Final simplification0.2

    \[\leadsto \left(4 \cdot y - 3 \cdot x\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \left(-z\right)\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6) (- (/ 2 3) z))))