Average Error: 26.0 → 0.5
Time: 28.2s
Precision: 64
\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.247417882216234 \cdot 10^{+41}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 3.1512810687298216 \cdot 10^{+66}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \end{array}\]
\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.247417882216234 \cdot 10^{+41}:\\
\;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\

\mathbf{elif}\;x \le 3.1512810687298216 \cdot 10^{+66}:\\
\;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\

\end{array}
double f(double x, double y, double z) {
        double r20551286 = x;
        double r20551287 = 2.0;
        double r20551288 = r20551286 - r20551287;
        double r20551289 = 4.16438922228;
        double r20551290 = r20551286 * r20551289;
        double r20551291 = 78.6994924154;
        double r20551292 = r20551290 + r20551291;
        double r20551293 = r20551292 * r20551286;
        double r20551294 = 137.519416416;
        double r20551295 = r20551293 + r20551294;
        double r20551296 = r20551295 * r20551286;
        double r20551297 = y;
        double r20551298 = r20551296 + r20551297;
        double r20551299 = r20551298 * r20551286;
        double r20551300 = z;
        double r20551301 = r20551299 + r20551300;
        double r20551302 = r20551288 * r20551301;
        double r20551303 = 43.3400022514;
        double r20551304 = r20551286 + r20551303;
        double r20551305 = r20551304 * r20551286;
        double r20551306 = 263.505074721;
        double r20551307 = r20551305 + r20551306;
        double r20551308 = r20551307 * r20551286;
        double r20551309 = 313.399215894;
        double r20551310 = r20551308 + r20551309;
        double r20551311 = r20551310 * r20551286;
        double r20551312 = 47.066876606;
        double r20551313 = r20551311 + r20551312;
        double r20551314 = r20551302 / r20551313;
        return r20551314;
}
↓
double f(double x, double y, double z) {
        double r20551315 = x;
        double r20551316 = -1.247417882216234e+41;
        bool r20551317 = r20551315 <= r20551316;
        double r20551318 = 4.16438922228;
        double r20551319 = r20551318 * r20551315;
        double r20551320 = y;
        double r20551321 = r20551315 * r20551315;
        double r20551322 = r20551320 / r20551321;
        double r20551323 = 110.1139242984811;
        double r20551324 = r20551322 - r20551323;
        double r20551325 = r20551319 + r20551324;
        double r20551326 = 3.1512810687298216e+66;
        bool r20551327 = r20551315 <= r20551326;
        double r20551328 = 2.0;
        double r20551329 = r20551315 - r20551328;
        double r20551330 = z;
        double r20551331 = 137.519416416;
        double r20551332 = 78.6994924154;
        double r20551333 = r20551319 + r20551332;
        double r20551334 = r20551315 * r20551333;
        double r20551335 = r20551331 + r20551334;
        double r20551336 = r20551315 * r20551335;
        double r20551337 = r20551320 + r20551336;
        double r20551338 = r20551315 * r20551337;
        double r20551339 = r20551330 + r20551338;
        double r20551340 = 47.066876606;
        double r20551341 = 313.399215894;
        double r20551342 = 263.505074721;
        double r20551343 = 43.3400022514;
        double r20551344 = r20551315 + r20551343;
        double r20551345 = r20551344 * r20551315;
        double r20551346 = r20551342 + r20551345;
        double r20551347 = r20551315 * r20551346;
        double r20551348 = r20551341 + r20551347;
        double r20551349 = r20551315 * r20551348;
        double r20551350 = r20551340 + r20551349;
        double r20551351 = r20551339 / r20551350;
        double r20551352 = r20551329 * r20551351;
        double r20551353 = r20551327 ? r20551352 : r20551325;
        double r20551354 = r20551317 ? r20551325 : r20551353;
        return r20551354;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.0
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.247417882216234e+41 or 3.1512810687298216e+66 < x

    1. Initial program 60.2

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

    2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922228 \cdot x\right) - 110.1139242984811}\]

    3. Simplified0.5

      \[\leadsto \color{blue}{4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)}\]

    if -1.247417882216234e+41 < x < 3.1512810687298216e+66

    1. Initial program 1.6

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity1.6

      \[\leadsto \frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606\right)}}\]

    4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}\]

    5. Simplified0.5

      \[\leadsto \color{blue}{\left(x - 2.0\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.247417882216234 \cdot 10^{+41}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 3.1512810687298216 \cdot 10^{+66}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))