Average Error: 26.0 → 0.8
Time: 22.7s
Precision: 64
\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.140336158145698 \cdot 10^{+17}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \le 1.692613215272597 \cdot 10^{+44}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \end{array}\]
\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 -5.140336158145698 \cdot 10^{+17}:\\
\;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\

\mathbf{elif}\;x \le 1.692613215272597 \cdot 10^{+44}:\\
\;\;\;\;\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}:\\
\;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r21223952 = x;
        double r21223953 = 2.0;
        double r21223954 = r21223952 - r21223953;
        double r21223955 = 4.16438922228;
        double r21223956 = r21223952 * r21223955;
        double r21223957 = 78.6994924154;
        double r21223958 = r21223956 + r21223957;
        double r21223959 = r21223958 * r21223952;
        double r21223960 = 137.519416416;
        double r21223961 = r21223959 + r21223960;
        double r21223962 = r21223961 * r21223952;
        double r21223963 = y;
        double r21223964 = r21223962 + r21223963;
        double r21223965 = r21223964 * r21223952;
        double r21223966 = z;
        double r21223967 = r21223965 + r21223966;
        double r21223968 = r21223954 * r21223967;
        double r21223969 = 43.3400022514;
        double r21223970 = r21223952 + r21223969;
        double r21223971 = r21223970 * r21223952;
        double r21223972 = 263.505074721;
        double r21223973 = r21223971 + r21223972;
        double r21223974 = r21223973 * r21223952;
        double r21223975 = 313.399215894;
        double r21223976 = r21223974 + r21223975;
        double r21223977 = r21223976 * r21223952;
        double r21223978 = 47.066876606;
        double r21223979 = r21223977 + r21223978;
        double r21223980 = r21223968 / r21223979;
        return r21223980;
}
↓
double f(double x, double y, double z) {
        double r21223981 = x;
        double r21223982 = -5.140336158145698e+17;
        bool r21223983 = r21223981 <= r21223982;
        double r21223984 = 4.16438922228;
        double r21223985 = r21223984 * r21223981;
        double r21223986 = 110.1139242984811;
        double r21223987 = r21223985 - r21223986;
        double r21223988 = y;
        double r21223989 = r21223988 / r21223981;
        double r21223990 = r21223989 / r21223981;
        double r21223991 = r21223987 + r21223990;
        double r21223992 = 1.692613215272597e+44;
        bool r21223993 = r21223981 <= r21223992;
        double r21223994 = 2.0;
        double r21223995 = r21223981 - r21223994;
        double r21223996 = z;
        double r21223997 = 137.519416416;
        double r21223998 = 78.6994924154;
        double r21223999 = r21223985 + r21223998;
        double r21224000 = r21223981 * r21223999;
        double r21224001 = r21223997 + r21224000;
        double r21224002 = r21223981 * r21224001;
        double r21224003 = r21223988 + r21224002;
        double r21224004 = r21223981 * r21224003;
        double r21224005 = r21223996 + r21224004;
        double r21224006 = 47.066876606;
        double r21224007 = 313.399215894;
        double r21224008 = 263.505074721;
        double r21224009 = 43.3400022514;
        double r21224010 = r21223981 + r21224009;
        double r21224011 = r21224010 * r21223981;
        double r21224012 = r21224008 + r21224011;
        double r21224013 = r21223981 * r21224012;
        double r21224014 = r21224007 + r21224013;
        double r21224015 = r21223981 * r21224014;
        double r21224016 = r21224006 + r21224015;
        double r21224017 = r21224005 / r21224016;
        double r21224018 = r21223995 * r21224017;
        double r21224019 = r21223993 ? r21224018 : r21223991;
        double r21224020 = r21223983 ? r21223991 : r21224019;
        return r21224020;
}

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
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.140336158145698e+17 or 1.692613215272597e+44 < x

    1. Initial program 56.2

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

    2. Taylor expanded around inf 1.3

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

    3. Simplified1.3

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

    if -5.140336158145698e+17 < x < 1.692613215272597e+44

    1. Initial program 0.7

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

    3. Applied *-un-lft-identity0.7

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.140336158145698 \cdot 10^{+17}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \le 1.692613215272597 \cdot 10^{+44}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))