Average Error: 0.4 → 0.3
Time: 4.8s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right) + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \left(\left(y - x\right) \cdot 6\right)
double f(double x, double y, double z) {
        double r262316 = x;
        double r262317 = y;
        double r262318 = r262317 - r262316;
        double r262319 = 6.0;
        double r262320 = r262318 * r262319;
        double r262321 = 2.0;
        double r262322 = 3.0;
        double r262323 = r262321 / r262322;
        double r262324 = z;
        double r262325 = r262323 - r262324;
        double r262326 = r262320 * r262325;
        double r262327 = r262316 + r262326;
        return r262327;
}
↓
double f(double x, double y, double z) {
        double r262328 = y;
        double r262329 = x;
        double r262330 = r262328 - r262329;
        double r262331 = 6.0;
        double r262332 = 2.0;
        double r262333 = 3.0;
        double r262334 = r262332 / r262333;
        double r262335 = z;
        double r262336 = r262334 - r262335;
        double r262337 = r262331 * r262336;
        double r262338 = fma(r262330, r262337, r262329);
        double r262339 = cbrt(r262335);
        double r262340 = -r262339;
        double r262341 = r262339 * r262339;
        double r262342 = r262339 * r262341;
        double r262343 = fma(r262340, r262341, r262342);
        double r262344 = r262330 * r262331;
        double r262345 = r262343 * r262344;
        double r262346 = r262338 + r262345;
        return r262346;
}

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 add-cube-cbrt0.8

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

  4. Applied add-sqr-sqrt0.8

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

  5. Applied prod-diff0.8

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

  6. Applied distribute-rgt-in0.8

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

  7. Applied associate-+r+0.8

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 +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))))