Average Error: 26.4 → 0.7
Time: 23.6s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10087778421912963069680247295975142927110000 \lor \neg \left(x \le 542737417535511941087232\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), y\right), x, z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), 47.06687660600000100430406746454536914825\right)} \cdot \left(x - 2\right)\\ \end{array}\]
\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 -10087778421912963069680247295975142927110000 \lor \neg \left(x \le 542737417535511941087232\right):\\
\;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r305879 = x;
        double r305880 = 2.0;
        double r305881 = r305879 - r305880;
        double r305882 = 4.16438922228;
        double r305883 = r305879 * r305882;
        double r305884 = 78.6994924154;
        double r305885 = r305883 + r305884;
        double r305886 = r305885 * r305879;
        double r305887 = 137.519416416;
        double r305888 = r305886 + r305887;
        double r305889 = r305888 * r305879;
        double r305890 = y;
        double r305891 = r305889 + r305890;
        double r305892 = r305891 * r305879;
        double r305893 = z;
        double r305894 = r305892 + r305893;
        double r305895 = r305881 * r305894;
        double r305896 = 43.3400022514;
        double r305897 = r305879 + r305896;
        double r305898 = r305897 * r305879;
        double r305899 = 263.505074721;
        double r305900 = r305898 + r305899;
        double r305901 = r305900 * r305879;
        double r305902 = 313.399215894;
        double r305903 = r305901 + r305902;
        double r305904 = r305903 * r305879;
        double r305905 = 47.066876606;
        double r305906 = r305904 + r305905;
        double r305907 = r305895 / r305906;
        return r305907;
}
↓
double f(double x, double y, double z) {
        double r305908 = x;
        double r305909 = -1.0087778421912963e+43;
        bool r305910 = r305908 <= r305909;
        double r305911 = 5.4273741753551194e+23;
        bool r305912 = r305908 <= r305911;
        double r305913 = !r305912;
        bool r305914 = r305910 || r305913;
        double r305915 = 4.16438922228;
        double r305916 = y;
        double r305917 = 2.0;
        double r305918 = pow(r305908, r305917);
        double r305919 = r305916 / r305918;
        double r305920 = 110.1139242984811;
        double r305921 = r305919 - r305920;
        double r305922 = fma(r305915, r305908, r305921);
        double r305923 = 78.6994924154;
        double r305924 = fma(r305915, r305908, r305923);
        double r305925 = 137.519416416;
        double r305926 = fma(r305924, r305908, r305925);
        double r305927 = fma(r305908, r305926, r305916);
        double r305928 = z;
        double r305929 = fma(r305927, r305908, r305928);
        double r305930 = 43.3400022514;
        double r305931 = r305908 + r305930;
        double r305932 = 263.505074721;
        double r305933 = fma(r305931, r305908, r305932);
        double r305934 = 313.399215894;
        double r305935 = fma(r305908, r305933, r305934);
        double r305936 = 47.066876606;
        double r305937 = fma(r305908, r305935, r305936);
        double r305938 = r305929 / r305937;
        double r305939 = 2.0;
        double r305940 = r305908 - r305939;
        double r305941 = r305938 * r305940;
        double r305942 = r305914 ? r305922 : r305941;
        return r305942;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.4
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0087778421912963e+43 or 5.4273741753551194e+23 < x

    1. Initial program 59.1

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

    2. Simplified54.9

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

    3. Taylor expanded around inf 1.2

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

    4. Simplified1.2

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

    if -1.0087778421912963e+43 < x < 5.4273741753551194e+23

    1. Initial program 0.7

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

    2. Simplified0.2

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

    4. Applied *-un-lft-identity0.2

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

    5. Applied *-un-lft-identity0.2

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

    6. Applied times-frac0.2

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

    7. Applied *-un-lft-identity0.2

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

    8. Applied times-frac0.2

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

    9. Simplified0.2

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

    10. Simplified0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10087778421912963069680247295975142927110000 \lor \neg \left(x \le 542737417535511941087232\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, 78.69949241540000173245061887428164482117\right), x, 137.5194164160000127594685181975364685059\right), y\right), x, z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), 313.3992158940000081202015280723571777344\right), 47.06687660600000100430406746454536914825\right)} \cdot \left(x - 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(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)))