Average Error: 27.1 → 0.5
Time: 7.0s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.010171170900404960706883500945882179821 \cdot 10^{65} \lor \neg \left(x \le 1.247264604620864180664669395148825741516 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}\right)\\ \end{array}\]
\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 -1.010171170900404960706883500945882179821 \cdot 10^{65} \lor \neg \left(x \le 1.247264604620864180664669395148825741516 \cdot 10^{55}\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x - 2\right) \cdot \left(1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r311874 = x;
        double r311875 = 2.0;
        double r311876 = r311874 - r311875;
        double r311877 = 4.16438922228;
        double r311878 = r311874 * r311877;
        double r311879 = 78.6994924154;
        double r311880 = r311878 + r311879;
        double r311881 = r311880 * r311874;
        double r311882 = 137.519416416;
        double r311883 = r311881 + r311882;
        double r311884 = r311883 * r311874;
        double r311885 = y;
        double r311886 = r311884 + r311885;
        double r311887 = r311886 * r311874;
        double r311888 = z;
        double r311889 = r311887 + r311888;
        double r311890 = r311876 * r311889;
        double r311891 = 43.3400022514;
        double r311892 = r311874 + r311891;
        double r311893 = r311892 * r311874;
        double r311894 = 263.505074721;
        double r311895 = r311893 + r311894;
        double r311896 = r311895 * r311874;
        double r311897 = 313.399215894;
        double r311898 = r311896 + r311897;
        double r311899 = r311898 * r311874;
        double r311900 = 47.066876606;
        double r311901 = r311899 + r311900;
        double r311902 = r311890 / r311901;
        return r311902;
}
↓
double f(double x, double y, double z) {
        double r311903 = x;
        double r311904 = -1.010171170900405e+65;
        bool r311905 = r311903 <= r311904;
        double r311906 = 1.2472646046208642e+55;
        bool r311907 = r311903 <= r311906;
        double r311908 = !r311907;
        bool r311909 = r311905 || r311908;
        double r311910 = 4.16438922228;
        double r311911 = y;
        double r311912 = 2.0;
        double r311913 = pow(r311903, r311912);
        double r311914 = r311911 / r311913;
        double r311915 = 110.1139242984811;
        double r311916 = r311914 - r311915;
        double r311917 = fma(r311903, r311910, r311916);
        double r311918 = 2.0;
        double r311919 = r311903 - r311918;
        double r311920 = 1.0;
        double r311921 = 78.6994924154;
        double r311922 = fma(r311903, r311910, r311921);
        double r311923 = 137.519416416;
        double r311924 = fma(r311922, r311903, r311923);
        double r311925 = fma(r311924, r311903, r311911);
        double r311926 = z;
        double r311927 = fma(r311925, r311903, r311926);
        double r311928 = 43.3400022514;
        double r311929 = r311903 + r311928;
        double r311930 = 263.505074721;
        double r311931 = fma(r311929, r311903, r311930);
        double r311932 = 313.399215894;
        double r311933 = fma(r311931, r311903, r311932);
        double r311934 = 47.066876606;
        double r311935 = fma(r311933, r311903, r311934);
        double r311936 = r311927 / r311935;
        double r311937 = r311920 * r311936;
        double r311938 = r311919 * r311937;
        double r311939 = r311909 ? r311917 : r311938;
        return r311939;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.1
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 < -1.010171170900405e+65 or 1.2472646046208642e+55 < x

    1. Initial program 63.5

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

    2. Simplified60.3

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

    if -1.010171170900405e+65 < x < 1.2472646046208642e+55

    1. Initial program 1.7

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

    2. Simplified0.9

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}}\]
    3. Using strategy rm

    4. Applied div-inv0.9

      \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.9

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

    7. Applied *-un-lft-identity0.9

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

    8. Applied times-frac0.9

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

    9. Applied add-cube-cbrt0.9

      \[\leadsto \left(x - 2\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\]

    10. Applied times-frac0.9

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\right)}\]

    11. Simplified0.9

      \[\leadsto \left(x - 2\right) \cdot \left(\color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}}\right)\]

    12. Simplified0.7

      \[\leadsto \left(x - 2\right) \cdot \left(1 \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.010171170900404960706883500945882179821 \cdot 10^{65} \lor \neg \left(x \le 1.247264604620864180664669395148825741516 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 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) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))