Data.Colour.CIE:cieLAB from colour-2.3.3, A

Average Error: 0.2 → 0.4

Time: 12.0s

Precision: 64

Math

\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]

↓

\[3 \cdot \left(\frac{-1}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}}} \cdot \frac{y}{\frac{\sqrt{\sqrt[3]{116}}}{\frac{16}{\sqrt{116}}}} + x \cdot y\right)\]

C

double f(double x, double y) {
        double r700144 = x;
        double r700145 = 16.0;
        double r700146 = 116.0;
        double r700147 = r700145 / r700146;
        double r700148 = r700144 - r700147;
        double r700149 = 3.0;
        double r700150 = r700148 * r700149;
        double r700151 = y;
        double r700152 = r700150 * r700151;
        return r700152;
}

↓

double f(double x, double y) {
        double r700153 = 3.0;
        double r700154 = -1.0;
        double r700155 = 116.0;
        double r700156 = cbrt(r700155);
        double r700157 = r700156 * r700156;
        double r700158 = sqrt(r700157);
        double r700159 = r700154 / r700158;
        double r700160 = y;
        double r700161 = sqrt(r700156);
        double r700162 = 16.0;
        double r700163 = sqrt(r700155);
        double r700164 = r700162 / r700163;
        double r700165 = r700161 / r700164;
        double r700166 = r700160 / r700165;
        double r700167 = r700159 * r700166;
        double r700168 = x;
        double r700169 = r700168 * r700160;
        double r700170 = r700167 + r700169;
        double r700171 = r700153 * r700170;
        return r700171;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.4

\[y \cdot \left(x \cdot 3 - 0.4137931034482758563264326312491903081536\right)\]

Derivation

1. Initial program 0.2

   \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]

2. Simplified 0.3

   \[\leadsto \color{blue}{\left(y \cdot \left(x - \frac{16}{116}\right)\right) \cdot 3}\]
3. Using strategy rm

4. Applied sub-neg 0.3

   \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-\frac{16}{116}\right)\right)}\right) \cdot 3\]

5. Applied distribute-lft-in 0.3

   \[\leadsto \color{blue}{\left(y \cdot x + y \cdot \left(-\frac{16}{116}\right)\right)} \cdot 3\]

6. Simplified 0.3

   \[\leadsto \left(\color{blue}{x \cdot y} + y \cdot \left(-\frac{16}{116}\right)\right) \cdot 3\]

7. Simplified 0.3

   \[\leadsto \left(x \cdot y + \color{blue}{\left(-y\right) \cdot \frac{16}{116}}\right) \cdot 3\]
8. Using strategy rm

9. Applied add-sqr-sqrt 0.7

   \[\leadsto \left(x \cdot y + \left(-y\right) \cdot \frac{16}{\color{blue}{\sqrt{116} \cdot \sqrt{116}}}\right) \cdot 3\]

10. Applied *-un-lft-identity 0.7

    \[\leadsto \left(x \cdot y + \left(-y\right) \cdot \frac{\color{blue}{1 \cdot 16}}{\sqrt{116} \cdot \sqrt{116}}\right) \cdot 3\]

11. Applied times-frac 0.3

    \[\leadsto \left(x \cdot y + \left(-y\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{116}} \cdot \frac{16}{\sqrt{116}}\right)}\right) \cdot 3\]

12. Applied associate-*r* 0.4

    \[\leadsto \left(x \cdot y + \color{blue}{\left(\left(-y\right) \cdot \frac{1}{\sqrt{116}}\right) \cdot \frac{16}{\sqrt{116}}}\right) \cdot 3\]

13. Simplified 0.4

    \[\leadsto \left(x \cdot y + \color{blue}{\frac{-y}{\sqrt{116}}} \cdot \frac{16}{\sqrt{116}}\right) \cdot 3\]
14. Using strategy rm

15. Applied add-cube-cbrt 0.4

    \[\leadsto \left(x \cdot y + \frac{-y}{\sqrt{\color{blue}{\left(\sqrt[3]{116} \cdot \sqrt[3]{116}\right) \cdot \sqrt[3]{116}}}} \cdot \frac{16}{\sqrt{116}}\right) \cdot 3\]

16. Applied sqrt-prod 0.4

    \[\leadsto \left(x \cdot y + \frac{-y}{\color{blue}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}} \cdot \sqrt{\sqrt[3]{116}}}} \cdot \frac{16}{\sqrt{116}}\right) \cdot 3\]

17. Applied *-un-lft-identity 0.4

    \[\leadsto \left(x \cdot y + \frac{-\color{blue}{1 \cdot y}}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}} \cdot \sqrt{\sqrt[3]{116}}} \cdot \frac{16}{\sqrt{116}}\right) \cdot 3\]

18. Applied distribute-lft-neg-in 0.4

    \[\leadsto \left(x \cdot y + \frac{\color{blue}{\left(-1\right) \cdot y}}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}} \cdot \sqrt{\sqrt[3]{116}}} \cdot \frac{16}{\sqrt{116}}\right) \cdot 3\]

19. Applied times-frac 0.4

    \[\leadsto \left(x \cdot y + \color{blue}{\left(\frac{-1}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}}} \cdot \frac{y}{\sqrt{\sqrt[3]{116}}}\right)} \cdot \frac{16}{\sqrt{116}}\right) \cdot 3\]

20. Applied associate-*l* 0.4

    \[\leadsto \left(x \cdot y + \color{blue}{\frac{-1}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}}} \cdot \left(\frac{y}{\sqrt{\sqrt[3]{116}}} \cdot \frac{16}{\sqrt{116}}\right)}\right) \cdot 3\]

21. Simplified 0.4

    \[\leadsto \left(x \cdot y + \frac{-1}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}}} \cdot \color{blue}{\frac{y}{\frac{\sqrt{\sqrt[3]{116}}}{\frac{16}{\sqrt{116}}}}}\right) \cdot 3\]

22. Final simplification 0.4

    \[\leadsto 3 \cdot \left(\frac{-1}{\sqrt{\sqrt[3]{116} \cdot \sqrt[3]{116}}} \cdot \frac{y}{\frac{\sqrt{\sqrt[3]{116}}}{\frac{16}{\sqrt{116}}}} + x \cdot y\right)\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"

  :herbie-target
  (* y (- (* x 3.0) 0.41379310344827586))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))