Average Error: 26.8 → 0.5
Time: 50.2s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -124741788221623399099317787892564199735300:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 3.151281068729821565811535199772462844528 \cdot 10^{66}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.5194164160000127594685181975364685059 + x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right)\right)\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]
\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 -124741788221623399099317787892564199735300:\\
\;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\

\mathbf{elif}\;x \le 3.151281068729821565811535199772462844528 \cdot 10^{66}:\\
\;\;\;\;\left(x - 2\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.5194164160000127594685181975364685059 + x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right)\right)\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r18728930 = x;
        double r18728931 = 2.0;
        double r18728932 = r18728930 - r18728931;
        double r18728933 = 4.16438922228;
        double r18728934 = r18728930 * r18728933;
        double r18728935 = 78.6994924154;
        double r18728936 = r18728934 + r18728935;
        double r18728937 = r18728936 * r18728930;
        double r18728938 = 137.519416416;
        double r18728939 = r18728937 + r18728938;
        double r18728940 = r18728939 * r18728930;
        double r18728941 = y;
        double r18728942 = r18728940 + r18728941;
        double r18728943 = r18728942 * r18728930;
        double r18728944 = z;
        double r18728945 = r18728943 + r18728944;
        double r18728946 = r18728932 * r18728945;
        double r18728947 = 43.3400022514;
        double r18728948 = r18728930 + r18728947;
        double r18728949 = r18728948 * r18728930;
        double r18728950 = 263.505074721;
        double r18728951 = r18728949 + r18728950;
        double r18728952 = r18728951 * r18728930;
        double r18728953 = 313.399215894;
        double r18728954 = r18728952 + r18728953;
        double r18728955 = r18728954 * r18728930;
        double r18728956 = 47.066876606;
        double r18728957 = r18728955 + r18728956;
        double r18728958 = r18728946 / r18728957;
        return r18728958;
}
↓
double f(double x, double y, double z) {
        double r18728959 = x;
        double r18728960 = -1.247417882216234e+41;
        bool r18728961 = r18728959 <= r18728960;
        double r18728962 = 4.16438922228;
        double r18728963 = r18728962 * r18728959;
        double r18728964 = y;
        double r18728965 = r18728959 * r18728959;
        double r18728966 = r18728964 / r18728965;
        double r18728967 = 110.1139242984811;
        double r18728968 = r18728966 - r18728967;
        double r18728969 = r18728963 + r18728968;
        double r18728970 = 3.1512810687298216e+66;
        bool r18728971 = r18728959 <= r18728970;
        double r18728972 = 2.0;
        double r18728973 = r18728959 - r18728972;
        double r18728974 = z;
        double r18728975 = 137.519416416;
        double r18728976 = 78.6994924154;
        double r18728977 = r18728963 + r18728976;
        double r18728978 = r18728959 * r18728977;
        double r18728979 = r18728975 + r18728978;
        double r18728980 = r18728959 * r18728979;
        double r18728981 = r18728964 + r18728980;
        double r18728982 = r18728959 * r18728981;
        double r18728983 = r18728974 + r18728982;
        double r18728984 = 47.066876606;
        double r18728985 = 313.399215894;
        double r18728986 = 263.505074721;
        double r18728987 = 43.3400022514;
        double r18728988 = r18728959 + r18728987;
        double r18728989 = r18728988 * r18728959;
        double r18728990 = r18728986 + r18728989;
        double r18728991 = r18728959 * r18728990;
        double r18728992 = r18728985 + r18728991;
        double r18728993 = r18728959 * r18728992;
        double r18728994 = r18728984 + r18728993;
        double r18728995 = r18728983 / r18728994;
        double r18728996 = r18728973 * r18728995;
        double r18728997 = r18728971 ? r18728996 : r18728969;
        double r18728998 = r18728961 ? r18728969 : r18728997;
        return r18728998;
}

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

Original26.8
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 62.2

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

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

    1. Initial program 1.6

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

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

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right)}}\]

    4. Applied times-frac0.6

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

    5. Simplified0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -124741788221623399099317787892564199735300:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 3.151281068729821565811535199772462844528 \cdot 10^{66}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.5194164160000127594685181975364685059 + x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right)\right)\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]

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) 1.0) (/ (+ (* (+ (* (+ (* (+ (* 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)))