Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 29.2 → 29.8

Time: 9.6s

Precision: 64

Math

\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

↓

\[\left(\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}\right) \cdot \frac{1 \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r64667 = x;
        double r64668 = y;
        double r64669 = r64667 * r64668;
        double r64670 = z;
        double r64671 = r64669 + r64670;
        double r64672 = r64671 * r64668;
        double r64673 = 27464.7644705;
        double r64674 = r64672 + r64673;
        double r64675 = r64674 * r64668;
        double r64676 = 230661.510616;
        double r64677 = r64675 + r64676;
        double r64678 = r64677 * r64668;
        double r64679 = t;
        double r64680 = r64678 + r64679;
        double r64681 = a;
        double r64682 = r64668 + r64681;
        double r64683 = r64682 * r64668;
        double r64684 = b;
        double r64685 = r64683 + r64684;
        double r64686 = r64685 * r64668;
        double r64687 = c;
        double r64688 = r64686 + r64687;
        double r64689 = r64688 * r64668;
        double r64690 = i;
        double r64691 = r64689 + r64690;
        double r64692 = r64680 / r64691;
        return r64692;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r64693 = x;
        double r64694 = y;
        double r64695 = r64693 * r64694;
        double r64696 = z;
        double r64697 = r64695 + r64696;
        double r64698 = r64697 * r64694;
        double r64699 = 27464.7644705;
        double r64700 = r64698 + r64699;
        double r64701 = r64700 * r64694;
        double r64702 = 230661.510616;
        double r64703 = r64701 + r64702;
        double r64704 = r64703 * r64694;
        double r64705 = t;
        double r64706 = r64704 + r64705;
        double r64707 = cbrt(r64706);
        double r64708 = r64707 * r64707;
        double r64709 = 1.0;
        double r64710 = r64709 * r64707;
        double r64711 = a;
        double r64712 = r64694 + r64711;
        double r64713 = b;
        double r64714 = fma(r64712, r64694, r64713);
        double r64715 = c;
        double r64716 = fma(r64714, r64694, r64715);
        double r64717 = i;
        double r64718 = fma(r64716, r64694, r64717);
        double r64719 = r64710 / r64718;
        double r64720 = r64708 * r64719;
        return r64720;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

1. Initial program 29.2

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
2. Using strategy rm

3. Applied clear-num 29.4

   \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}}\]
4. Using strategy rm

5. Applied add-cube-cbrt 29.9

   \[\leadsto \frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\color{blue}{\left(\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}\right) \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}}}\]

6. Applied *-un-lft-identity 29.9

   \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)}}{\left(\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}\right) \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}}\]

7. Applied times-frac 29.9

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}} \cdot \frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}}}\]

8. Applied add-cube-cbrt 29.9

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}} \cdot \frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}}\]

9. Applied times-frac 29.8

   \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}} \cdot \frac{\sqrt[3]{1}}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}}}\]

10. Simplified 29.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}\right)} \cdot \frac{\sqrt[3]{1}}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}}\]

11. Simplified 29.8

    \[\leadsto \left(\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}\right) \cdot \color{blue}{\frac{1 \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]

12. Final simplification 29.8

    \[\leadsto \left(\sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t} \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}\right) \cdot \frac{1 \cdot \sqrt[3]{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))