Average Error: 0.4 → 0.1
Time: 3.0s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(1, 4, -z \cdot 6\right), y - x, \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(\mathsf{fma}\left(1, 4, -z \cdot 6\right), y - x, \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\right)
double f(double x, double y, double z) {
        double r228573 = x;
        double r228574 = y;
        double r228575 = r228574 - r228573;
        double r228576 = 6.0;
        double r228577 = r228575 * r228576;
        double r228578 = 2.0;
        double r228579 = 3.0;
        double r228580 = r228578 / r228579;
        double r228581 = z;
        double r228582 = r228580 - r228581;
        double r228583 = r228577 * r228582;
        double r228584 = r228573 + r228583;
        return r228584;
}
↓
double f(double x, double y, double z) {
        double r228585 = 1.0;
        double r228586 = 4.0;
        double r228587 = z;
        double r228588 = 6.0;
        double r228589 = r228587 * r228588;
        double r228590 = -r228589;
        double r228591 = fma(r228585, r228586, r228590);
        double r228592 = y;
        double r228593 = x;
        double r228594 = r228592 - r228593;
        double r228595 = -r228587;
        double r228596 = fma(r228595, r228588, r228589);
        double r228597 = fma(r228596, r228594, r228593);
        double r228598 = fma(r228591, r228594, r228597);
        return r228598;
}

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. Simplified0.2

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4 - 6 \cdot z}, x\right)\]
  4. Using strategy rm

  5. Applied fma-udef0.2

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

  7. Applied *-un-lft-identity0.2

    \[\leadsto \left(y - x\right) \cdot \left(\color{blue}{1 \cdot 4} - 6 \cdot z\right) + x\]

  8. Applied prod-diff0.2

    \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\mathsf{fma}\left(1, 4, -z \cdot 6\right) + \mathsf{fma}\left(-z, 6, z \cdot 6\right)\right)} + x\]

  9. Applied distribute-rgt-in0.2

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(1, 4, -z \cdot 6\right) \cdot \left(y - x\right) + \mathsf{fma}\left(-z, 6, z \cdot 6\right) \cdot \left(y - x\right)\right)} + x\]

  10. Applied associate-+l+0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, 4, -z \cdot 6\right) \cdot \left(y - x\right) + \left(\mathsf{fma}\left(-z, 6, z \cdot 6\right) \cdot \left(y - x\right) + x\right)}\]

  11. Simplified0.2

    \[\leadsto \mathsf{fma}\left(1, 4, -z \cdot 6\right) \cdot \left(y - x\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)}\]
  12. Using strategy rm

  13. Applied fma-def0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1, 4, -z \cdot 6\right), y - x, \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\right)}\]

  14. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1, 4, -z \cdot 6\right), y - x, \mathsf{fma}\left(\mathsf{fma}\left(-z, 6, z \cdot 6\right), y - x, x\right)\right)\]

Reproduce

herbie shell --seed 2020056 +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))))