Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 29.2 → 29.8

Time: 10.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r63265 = x;
        double r63266 = y;
        double r63267 = r63265 * r63266;
        double r63268 = z;
        double r63269 = r63267 + r63268;
        double r63270 = r63269 * r63266;
        double r63271 = 27464.7644705;
        double r63272 = r63270 + r63271;
        double r63273 = r63272 * r63266;
        double r63274 = 230661.510616;
        double r63275 = r63273 + r63274;
        double r63276 = r63275 * r63266;
        double r63277 = t;
        double r63278 = r63276 + r63277;
        double r63279 = a;
        double r63280 = r63266 + r63279;
        double r63281 = r63280 * r63266;
        double r63282 = b;
        double r63283 = r63281 + r63282;
        double r63284 = r63283 * r63266;
        double r63285 = c;
        double r63286 = r63284 + r63285;
        double r63287 = r63286 * r63266;
        double r63288 = i;
        double r63289 = r63287 + r63288;
        double r63290 = r63278 / r63289;
        return r63290;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r63291 = x;
        double r63292 = y;
        double r63293 = r63291 * r63292;
        double r63294 = z;
        double r63295 = r63293 + r63294;
        double r63296 = r63295 * r63292;
        double r63297 = 27464.7644705;
        double r63298 = r63296 + r63297;
        double r63299 = r63298 * r63292;
        double r63300 = 230661.510616;
        double r63301 = r63299 + r63300;
        double r63302 = r63301 * r63292;
        double r63303 = t;
        double r63304 = r63302 + r63303;
        double r63305 = cbrt(r63304);
        double r63306 = r63305 * r63305;
        double r63307 = 1.0;
        double r63308 = r63307 * r63305;
        double r63309 = a;
        double r63310 = r63292 + r63309;
        double r63311 = b;
        double r63312 = fma(r63310, r63292, r63311);
        double r63313 = c;
        double r63314 = fma(r63312, r63292, r63313);
        double r63315 = i;
        double r63316 = fma(r63314, r63292, r63315);
        double r63317 = r63308 / r63316;
        double r63318 = r63306 * r63317;
        return r63318;
}

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)))