Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.8 → 28.9

Time: 33.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r49986 = x;
        double r49987 = y;
        double r49988 = r49986 * r49987;
        double r49989 = z;
        double r49990 = r49988 + r49989;
        double r49991 = r49990 * r49987;
        double r49992 = 27464.7644705;
        double r49993 = r49991 + r49992;
        double r49994 = r49993 * r49987;
        double r49995 = 230661.510616;
        double r49996 = r49994 + r49995;
        double r49997 = r49996 * r49987;
        double r49998 = t;
        double r49999 = r49997 + r49998;
        double r50000 = a;
        double r50001 = r49987 + r50000;
        double r50002 = r50001 * r49987;
        double r50003 = b;
        double r50004 = r50002 + r50003;
        double r50005 = r50004 * r49987;
        double r50006 = c;
        double r50007 = r50005 + r50006;
        double r50008 = r50007 * r49987;
        double r50009 = i;
        double r50010 = r50008 + r50009;
        double r50011 = r49999 / r50010;
        return r50011;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r50012 = x;
        double r50013 = y;
        double r50014 = z;
        double r50015 = fma(r50012, r50013, r50014);
        double r50016 = 27464.7644705;
        double r50017 = fma(r50015, r50013, r50016);
        double r50018 = 230661.510616;
        double r50019 = fma(r50017, r50013, r50018);
        double r50020 = cbrt(r50019);
        double r50021 = r50020 * r50020;
        double r50022 = r50021 * r50020;
        double r50023 = t;
        double r50024 = fma(r50022, r50013, r50023);
        double r50025 = a;
        double r50026 = r50013 + r50025;
        double r50027 = b;
        double r50028 = fma(r50026, r50013, r50027);
        double r50029 = c;
        double r50030 = fma(r50028, r50013, r50029);
        double r50031 = i;
        double r50032 = fma(r50030, r50013, r50031);
        double r50033 = r50024 / r50032;
        return r50033;
}

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 28.8

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

2. Simplified 28.8

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

4. Applied add-cube-cbrt 28.9

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

5. Final simplification 28.9

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

Reproduce

herbie shell --seed 2019303 +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) 27464.764470499998) y) 230661.510616000014) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))