Average Error: 26.1 → 1.1
Time: 24.2s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2252101360098186826350592 \lor \neg \left(x \le 14456844266184348924014231552\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\sqrt{\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059} \cdot \left(\sqrt{\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059} \cdot x\right) + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]
\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 -2252101360098186826350592 \lor \neg \left(x \le 14456844266184348924014231552\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\sqrt{\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059} \cdot \left(\sqrt{\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059} \cdot x\right) + 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}\\

\end{array}
double f(double x, double y, double z) {
        double r940836 = x;
        double r940837 = 2.0;
        double r940838 = r940836 - r940837;
        double r940839 = 4.16438922228;
        double r940840 = r940836 * r940839;
        double r940841 = 78.6994924154;
        double r940842 = r940840 + r940841;
        double r940843 = r940842 * r940836;
        double r940844 = 137.519416416;
        double r940845 = r940843 + r940844;
        double r940846 = r940845 * r940836;
        double r940847 = y;
        double r940848 = r940846 + r940847;
        double r940849 = r940848 * r940836;
        double r940850 = z;
        double r940851 = r940849 + r940850;
        double r940852 = r940838 * r940851;
        double r940853 = 43.3400022514;
        double r940854 = r940836 + r940853;
        double r940855 = r940854 * r940836;
        double r940856 = 263.505074721;
        double r940857 = r940855 + r940856;
        double r940858 = r940857 * r940836;
        double r940859 = 313.399215894;
        double r940860 = r940858 + r940859;
        double r940861 = r940860 * r940836;
        double r940862 = 47.066876606;
        double r940863 = r940861 + r940862;
        double r940864 = r940852 / r940863;
        return r940864;
}
↓
double f(double x, double y, double z) {
        double r940865 = x;
        double r940866 = -2.2521013600981868e+24;
        bool r940867 = r940865 <= r940866;
        double r940868 = 1.445684426618435e+28;
        bool r940869 = r940865 <= r940868;
        double r940870 = !r940869;
        bool r940871 = r940867 || r940870;
        double r940872 = y;
        double r940873 = 2.0;
        double r940874 = pow(r940865, r940873);
        double r940875 = r940872 / r940874;
        double r940876 = 4.16438922228;
        double r940877 = r940876 * r940865;
        double r940878 = r940875 + r940877;
        double r940879 = 110.1139242984811;
        double r940880 = r940878 - r940879;
        double r940881 = 2.0;
        double r940882 = r940865 - r940881;
        double r940883 = r940865 * r940876;
        double r940884 = 78.6994924154;
        double r940885 = r940883 + r940884;
        double r940886 = r940885 * r940865;
        double r940887 = 137.519416416;
        double r940888 = r940886 + r940887;
        double r940889 = sqrt(r940888);
        double r940890 = r940889 * r940865;
        double r940891 = r940889 * r940890;
        double r940892 = r940891 + r940872;
        double r940893 = r940892 * r940865;
        double r940894 = z;
        double r940895 = r940893 + r940894;
        double r940896 = r940882 * r940895;
        double r940897 = 43.3400022514;
        double r940898 = r940865 + r940897;
        double r940899 = r940898 * r940865;
        double r940900 = 263.505074721;
        double r940901 = r940899 + r940900;
        double r940902 = r940901 * r940865;
        double r940903 = 313.399215894;
        double r940904 = r940902 + r940903;
        double r940905 = r940904 * r940865;
        double r940906 = 47.066876606;
        double r940907 = r940905 + r940906;
        double r940908 = r940896 / r940907;
        double r940909 = r940871 ? r940880 : r940908;
        return r940909;
}

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

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

Derivation

  1. Split input into 2 regimes

  2. if x < -2.2521013600981868e+24 or 1.445684426618435e+28 < x

    1. Initial program 57.5

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

    2. Taylor expanded around inf 1.6

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

    if -2.2521013600981868e+24 < x < 1.445684426618435e+28

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt0.7

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

    4. Applied associate-*l*0.7

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2252101360098186826350592 \lor \neg \left(x \le 14456844266184348924014231552\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\sqrt{\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059} \cdot \left(\sqrt{\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059} \cdot x\right) + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.3261287258700048e62) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109) (if (< x 9.4299917145546727e55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z) (+ (* (+ (+ (* 263.50507472100003 x) (+ (* 43.3400022514000014 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606000001))) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514000014) x) 263.50507472100003) x) 313.399215894) x) 47.066876606000001)))