Average Error: 0.4 → 0.2
Time: 30.9s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(-z, 1, z\right) \cdot \left(\left(y - x\right) \cdot 6\right) + \mathsf{fma}\left(y - x, \left(\frac{\frac{2}{\sqrt{3}}}{\sqrt{3}} - z\right) \cdot 6, x\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(-z, 1, z\right) \cdot \left(\left(y - x\right) \cdot 6\right) + \mathsf{fma}\left(y - x, \left(\frac{\frac{2}{\sqrt{3}}}{\sqrt{3}} - z\right) \cdot 6, x\right)
double f(double x, double y, double z) {
        double r17244198 = x;
        double r17244199 = y;
        double r17244200 = r17244199 - r17244198;
        double r17244201 = 6.0;
        double r17244202 = r17244200 * r17244201;
        double r17244203 = 2.0;
        double r17244204 = 3.0;
        double r17244205 = r17244203 / r17244204;
        double r17244206 = z;
        double r17244207 = r17244205 - r17244206;
        double r17244208 = r17244202 * r17244207;
        double r17244209 = r17244198 + r17244208;
        return r17244209;
}
↓
double f(double x, double y, double z) {
        double r17244210 = z;
        double r17244211 = -r17244210;
        double r17244212 = 1.0;
        double r17244213 = fma(r17244211, r17244212, r17244210);
        double r17244214 = y;
        double r17244215 = x;
        double r17244216 = r17244214 - r17244215;
        double r17244217 = 6.0;
        double r17244218 = r17244216 * r17244217;
        double r17244219 = r17244213 * r17244218;
        double r17244220 = 2.0;
        double r17244221 = 3.0;
        double r17244222 = sqrt(r17244221);
        double r17244223 = r17244220 / r17244222;
        double r17244224 = r17244223 / r17244222;
        double r17244225 = r17244224 - r17244210;
        double r17244226 = r17244225 * r17244217;
        double r17244227 = fma(r17244216, r17244226, r17244215);
        double r17244228 = r17244219 + r17244227;
        return r17244228;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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 *-un-lft-identity0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\]

  4. Applied add-sqr-sqrt0.7

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}} - 1 \cdot z\right)\]

  5. Applied *-un-lft-identity0.7

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{\color{blue}{1 \cdot 2}}{\sqrt{3} \cdot \sqrt{3}} - 1 \cdot z\right)\]

  6. Applied times-frac1.1

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\color{blue}{\frac{1}{\sqrt{3}} \cdot \frac{2}{\sqrt{3}}} - 1 \cdot z\right)\]

  7. Applied prod-diff1.1

    \[\leadsto x + \left(\left(y - x\right) \cdot 6\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}, -z \cdot 1\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  8. Applied distribute-lft-in1.1

    \[\leadsto x + \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}, -z \cdot 1\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  9. Applied associate-+r+1.1

    \[\leadsto \color{blue}{\left(x + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}, -z \cdot 1\right)\right) + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)}\]

  10. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{\frac{2}{\sqrt{3}}}{\sqrt{3}} - z\right), x\right)} + \left(\left(y - x\right) \cdot 6\right) \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\]

  11. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(-z, 1, z\right) \cdot \left(\left(y - x\right) \cdot 6\right) + \mathsf{fma}\left(y - x, \left(\frac{\frac{2}{\sqrt{3}}}{\sqrt{3}} - z\right) \cdot 6, x\right)\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))