Average Error: 26.9 → 1.2
Time: 28.0s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -63074748513986640340139900928:\\ \;\;\;\;\left(x \cdot 4.16438922227999963610045597306452691555 + \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \le 620615997107907.625:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117 \cdot x\right) + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 4.16438922227999963610045597306452691555 + \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}
↓
\begin{array}{l}
\mathbf{if}\;x \le -63074748513986640340139900928:\\
\;\;\;\;\left(x \cdot 4.16438922227999963610045597306452691555 + \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\

\mathbf{elif}\;x \le 620615997107907.625:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117 \cdot x\right) + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot 4.16438922227999963610045597306452691555 + \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\

\end{array}
double f(double x, double y, double z) {
        double r21146850 = x;
        double r21146851 = 2.0;
        double r21146852 = r21146850 - r21146851;
        double r21146853 = 4.16438922228;
        double r21146854 = r21146850 * r21146853;
        double r21146855 = 78.6994924154;
        double r21146856 = r21146854 + r21146855;
        double r21146857 = r21146856 * r21146850;
        double r21146858 = 137.519416416;
        double r21146859 = r21146857 + r21146858;
        double r21146860 = r21146859 * r21146850;
        double r21146861 = y;
        double r21146862 = r21146860 + r21146861;
        double r21146863 = r21146862 * r21146850;
        double r21146864 = z;
        double r21146865 = r21146863 + r21146864;
        double r21146866 = r21146852 * r21146865;
        double r21146867 = 43.3400022514;
        double r21146868 = r21146850 + r21146867;
        double r21146869 = r21146868 * r21146850;
        double r21146870 = 263.505074721;
        double r21146871 = r21146869 + r21146870;
        double r21146872 = r21146871 * r21146850;
        double r21146873 = 313.399215894;
        double r21146874 = r21146872 + r21146873;
        double r21146875 = r21146874 * r21146850;
        double r21146876 = 47.066876606;
        double r21146877 = r21146875 + r21146876;
        double r21146878 = r21146866 / r21146877;
        return r21146878;
}
↓
double f(double x, double y, double z) {
        double r21146879 = x;
        double r21146880 = -6.307474851398664e+28;
        bool r21146881 = r21146879 <= r21146880;
        double r21146882 = 4.16438922228;
        double r21146883 = r21146879 * r21146882;
        double r21146884 = y;
        double r21146885 = r21146879 * r21146879;
        double r21146886 = r21146884 / r21146885;
        double r21146887 = r21146883 + r21146886;
        double r21146888 = 110.1139242984811;
        double r21146889 = r21146887 - r21146888;
        double r21146890 = 620615997107907.6;
        bool r21146891 = r21146879 <= r21146890;
        double r21146892 = 2.0;
        double r21146893 = r21146879 - r21146892;
        double r21146894 = r21146885 * r21146882;
        double r21146895 = 78.6994924154;
        double r21146896 = r21146895 * r21146879;
        double r21146897 = r21146894 + r21146896;
        double r21146898 = 137.519416416;
        double r21146899 = r21146897 + r21146898;
        double r21146900 = r21146899 * r21146879;
        double r21146901 = r21146900 + r21146884;
        double r21146902 = r21146901 * r21146879;
        double r21146903 = z;
        double r21146904 = r21146902 + r21146903;
        double r21146905 = r21146893 * r21146904;
        double r21146906 = 43.3400022514;
        double r21146907 = r21146879 + r21146906;
        double r21146908 = r21146907 * r21146879;
        double r21146909 = 263.505074721;
        double r21146910 = r21146908 + r21146909;
        double r21146911 = r21146910 * r21146879;
        double r21146912 = 313.399215894;
        double r21146913 = r21146911 + r21146912;
        double r21146914 = r21146913 * r21146879;
        double r21146915 = 47.066876606;
        double r21146916 = r21146914 + r21146915;
        double r21146917 = r21146905 / r21146916;
        double r21146918 = r21146891 ? r21146917 : r21146889;
        double r21146919 = r21146881 ? r21146889 : r21146918;
        return r21146919;
}

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

Original26.9
Target0.6
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6.307474851398664e+28 or 620615997107907.6 < x

    1. Initial program 56.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Taylor expanded around inf 2.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

    3. Simplified2.1

      \[\leadsto \color{blue}{\left(x \cdot 4.16438922227999963610045597306452691555 + \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229}\]

    if -6.307474851398664e+28 < x < 620615997107907.6

    1. Initial program 0.5

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Taylor expanded around 0 0.5

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\left(78.69949241540000173245061887428164482117 \cdot x + 4.16438922227999963610045597306452691555 \cdot {x}^{2}\right)} + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    3. Simplified0.5

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117 \cdot x\right)} + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -63074748513986640340139900928:\\ \;\;\;\;\left(x \cdot 4.16438922227999963610045597306452691555 + \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \le 620615997107907.625:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117 \cdot x\right) + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 4.16438922227999963610045597306452691555 + \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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) 1.0) (/ (+ (* (+ (* (+ (* (+ (* 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)))