Average Error: 29.0 → 28.1
Time: 9.6s
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 r73281 = x;
        double r73282 = y;
        double r73283 = r73281 * r73282;
        double r73284 = z;
        double r73285 = r73283 + r73284;
        double r73286 = r73285 * r73282;
        double r73287 = 27464.7644705;
        double r73288 = r73286 + r73287;
        double r73289 = r73288 * r73282;
        double r73290 = 230661.510616;
        double r73291 = r73289 + r73290;
        double r73292 = r73291 * r73282;
        double r73293 = t;
        double r73294 = r73292 + r73293;
        double r73295 = a;
        double r73296 = r73282 + r73295;
        double r73297 = r73296 * r73282;
        double r73298 = b;
        double r73299 = r73297 + r73298;
        double r73300 = r73299 * r73282;
        double r73301 = c;
        double r73302 = r73300 + r73301;
        double r73303 = r73302 * r73282;
        double r73304 = i;
        double r73305 = r73303 + r73304;
        double r73306 = r73294 / r73305;
        return r73306;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r73307 = x;
        double r73308 = y;
        double r73309 = r73307 * r73308;
        double r73310 = z;
        double r73311 = r73309 + r73310;
        double r73312 = r73311 * r73308;
        double r73313 = 27464.7644705;
        double r73314 = r73312 + r73313;
        double r73315 = r73314 * r73308;
        double r73316 = 230661.510616;
        double r73317 = r73315 + r73316;
        double r73318 = r73317 * r73308;
        double r73319 = t;
        double r73320 = r73318 + r73319;
        double r73321 = a;
        double r73322 = r73308 + r73321;
        double r73323 = r73322 * r73308;
        double r73324 = b;
        double r73325 = r73323 + r73324;
        double r73326 = r73325 * r73308;
        double r73327 = c;
        double r73328 = r73326 + r73327;
        double r73329 = r73328 * r73308;
        double r73330 = i;
        double r73331 = r73329 + r73330;
        double r73332 = r73320 / r73331;
        double r73333 = 5.142889239312206e+306;
        bool r73334 = r73332 <= r73333;
        double r73335 = 1.0;
        double r73336 = fma(r73322, r73308, r73324);
        double r73337 = fma(r73336, r73308, r73327);
        double r73338 = fma(r73337, r73308, r73330);
        double r73339 = r73338 * r73335;
        double r73340 = r73335 / r73339;
        double r73341 = r73320 * r73340;
        double r73342 = 0.0;
        double r73343 = r73334 ? r73341 : r73342;
        return r73343;
}

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)))