Average Error: 29.0 → 28.1
Time: 10.5s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 5.142889239312206 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 5.142889239312206 \cdot 10^{306}:\\
\;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r83285 = x;
        double r83286 = y;
        double r83287 = r83285 * r83286;
        double r83288 = z;
        double r83289 = r83287 + r83288;
        double r83290 = r83289 * r83286;
        double r83291 = 27464.7644705;
        double r83292 = r83290 + r83291;
        double r83293 = r83292 * r83286;
        double r83294 = 230661.510616;
        double r83295 = r83293 + r83294;
        double r83296 = r83295 * r83286;
        double r83297 = t;
        double r83298 = r83296 + r83297;
        double r83299 = a;
        double r83300 = r83286 + r83299;
        double r83301 = r83300 * r83286;
        double r83302 = b;
        double r83303 = r83301 + r83302;
        double r83304 = r83303 * r83286;
        double r83305 = c;
        double r83306 = r83304 + r83305;
        double r83307 = r83306 * r83286;
        double r83308 = i;
        double r83309 = r83307 + r83308;
        double r83310 = r83298 / r83309;
        return r83310;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r83311 = x;
        double r83312 = y;
        double r83313 = r83311 * r83312;
        double r83314 = z;
        double r83315 = r83313 + r83314;
        double r83316 = r83315 * r83312;
        double r83317 = 27464.7644705;
        double r83318 = r83316 + r83317;
        double r83319 = r83318 * r83312;
        double r83320 = 230661.510616;
        double r83321 = r83319 + r83320;
        double r83322 = r83321 * r83312;
        double r83323 = t;
        double r83324 = r83322 + r83323;
        double r83325 = a;
        double r83326 = r83312 + r83325;
        double r83327 = r83326 * r83312;
        double r83328 = b;
        double r83329 = r83327 + r83328;
        double r83330 = r83329 * r83312;
        double r83331 = c;
        double r83332 = r83330 + r83331;
        double r83333 = r83332 * r83312;
        double r83334 = i;
        double r83335 = r83333 + r83334;
        double r83336 = r83324 / r83335;
        double r83337 = 5.142889239312206e+306;
        bool r83338 = r83336 <= r83337;
        double r83339 = 1.0;
        double r83340 = fma(r83326, r83312, r83328);
        double r83341 = fma(r83340, r83312, r83331);
        double r83342 = fma(r83341, r83312, r83334);
        double r83343 = r83342 * r83339;
        double r83344 = r83339 / r83343;
        double r83345 = r83324 * r83344;
        double r83346 = 0.0;
        double r83347 = r83338 ? r83345 : r83346;
        return r83347;
}

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. Split input into 2 regimes

  2. if (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)) < 5.142889239312206e+306

    1. Initial program 5.2

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
    2. Using strategy rm

    3. Applied div-inv5.4

      \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}\]

    4. Simplified5.3

      \[\leadsto \left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}}\]

    if 5.142889239312206e+306 < (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))

    1. Initial program 64.0

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

    2. Taylor expanded around 0 61.8

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification28.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 5.142889239312206 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))