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 r73169 = x;
        double r73170 = y;
        double r73171 = r73169 * r73170;
        double r73172 = z;
        double r73173 = r73171 + r73172;
        double r73174 = r73173 * r73170;
        double r73175 = 27464.7644705;
        double r73176 = r73174 + r73175;
        double r73177 = r73176 * r73170;
        double r73178 = 230661.510616;
        double r73179 = r73177 + r73178;
        double r73180 = r73179 * r73170;
        double r73181 = t;
        double r73182 = r73180 + r73181;
        double r73183 = a;
        double r73184 = r73170 + r73183;
        double r73185 = r73184 * r73170;
        double r73186 = b;
        double r73187 = r73185 + r73186;
        double r73188 = r73187 * r73170;
        double r73189 = c;
        double r73190 = r73188 + r73189;
        double r73191 = r73190 * r73170;
        double r73192 = i;
        double r73193 = r73191 + r73192;
        double r73194 = r73182 / r73193;
        return r73194;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r73195 = x;
        double r73196 = y;
        double r73197 = r73195 * r73196;
        double r73198 = z;
        double r73199 = r73197 + r73198;
        double r73200 = r73199 * r73196;
        double r73201 = 27464.7644705;
        double r73202 = r73200 + r73201;
        double r73203 = r73202 * r73196;
        double r73204 = 230661.510616;
        double r73205 = r73203 + r73204;
        double r73206 = r73205 * r73196;
        double r73207 = t;
        double r73208 = r73206 + r73207;
        double r73209 = a;
        double r73210 = r73196 + r73209;
        double r73211 = r73210 * r73196;
        double r73212 = b;
        double r73213 = r73211 + r73212;
        double r73214 = r73213 * r73196;
        double r73215 = c;
        double r73216 = r73214 + r73215;
        double r73217 = r73216 * r73196;
        double r73218 = i;
        double r73219 = r73217 + r73218;
        double r73220 = r73208 / r73219;
        double r73221 = 5.142889239312206e+306;
        bool r73222 = r73220 <= r73221;
        double r73223 = 1.0;
        double r73224 = fma(r73210, r73196, r73212);
        double r73225 = fma(r73224, r73196, r73215);
        double r73226 = fma(r73225, r73196, r73218);
        double r73227 = r73226 * r73223;
        double r73228 = r73223 / r73227;
        double r73229 = r73208 * r73228;
        double r73230 = 0.0;
        double r73231 = r73222 ? r73229 : r73230;
        return r73231;
}

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)))