Average Error: 0.4 → 0.6
Time: 21.9s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right), \sqrt[3]{x}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right), \sqrt[3]{x}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\right)
double f(double x, double y, double z) {
        double r203467 = x;
        double r203468 = y;
        double r203469 = r203468 - r203467;
        double r203470 = 6.0;
        double r203471 = r203469 * r203470;
        double r203472 = 2.0;
        double r203473 = 3.0;
        double r203474 = r203472 / r203473;
        double r203475 = z;
        double r203476 = r203474 - r203475;
        double r203477 = r203471 * r203476;
        double r203478 = r203467 + r203477;
        return r203478;
}
↓
double f(double x, double y, double z) {
        double r203479 = x;
        double r203480 = cbrt(r203479);
        double r203481 = cbrt(r203480);
        double r203482 = r203480 * r203480;
        double r203483 = cbrt(r203482);
        double r203484 = cbrt(r203483);
        double r203485 = cbrt(r203481);
        double r203486 = r203484 * r203485;
        double r203487 = r203481 * r203486;
        double r203488 = r203487 * r203481;
        double r203489 = r203480 * r203488;
        double r203490 = y;
        double r203491 = r203490 - r203479;
        double r203492 = 6.0;
        double r203493 = r203491 * r203492;
        double r203494 = 2.0;
        double r203495 = 3.0;
        double r203496 = r203494 / r203495;
        double r203497 = z;
        double r203498 = r203496 - r203497;
        double r203499 = r203493 * r203498;
        double r203500 = fma(r203489, r203480, r203499);
        return r203500;
}

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-cbrt0.5

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

  4. Applied fma-def0.5

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

  6. Applied add-cube-cbrt0.6

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

  8. Applied add-cube-cbrt0.6

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

  9. Applied cbrt-prod0.6

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

  10. Applied cbrt-prod0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019304 +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))))