Average Error: 26.6 → 1.0
Time: 48.5s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -909753830361948524383850987520:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{\frac{y}{x}}{x}\right) - 110.1139242984810806547102401964366436005\\ \mathbf{elif}\;x \le 1498395052788.9189453125:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x \cdot x - 2 \cdot 2} \cdot \left(x + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]
\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 -909753830361948524383850987520:\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{\frac{y}{x}}{x}\right) - 110.1139242984810806547102401964366436005\\

\mathbf{elif}\;x \le 1498395052788.9189453125:\\
\;\;\;\;\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(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x \cdot x - 2 \cdot 2} \cdot \left(x + 2\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r15031934 = x;
        double r15031935 = 2.0;
        double r15031936 = r15031934 - r15031935;
        double r15031937 = 4.16438922228;
        double r15031938 = r15031934 * r15031937;
        double r15031939 = 78.6994924154;
        double r15031940 = r15031938 + r15031939;
        double r15031941 = r15031940 * r15031934;
        double r15031942 = 137.519416416;
        double r15031943 = r15031941 + r15031942;
        double r15031944 = r15031943 * r15031934;
        double r15031945 = y;
        double r15031946 = r15031944 + r15031945;
        double r15031947 = r15031946 * r15031934;
        double r15031948 = z;
        double r15031949 = r15031947 + r15031948;
        double r15031950 = r15031936 * r15031949;
        double r15031951 = 43.3400022514;
        double r15031952 = r15031934 + r15031951;
        double r15031953 = r15031952 * r15031934;
        double r15031954 = 263.505074721;
        double r15031955 = r15031953 + r15031954;
        double r15031956 = r15031955 * r15031934;
        double r15031957 = 313.399215894;
        double r15031958 = r15031956 + r15031957;
        double r15031959 = r15031958 * r15031934;
        double r15031960 = 47.066876606;
        double r15031961 = r15031959 + r15031960;
        double r15031962 = r15031950 / r15031961;
        return r15031962;
}
↓
double f(double x, double y, double z) {
        double r15031963 = x;
        double r15031964 = -9.097538303619485e+29;
        bool r15031965 = r15031963 <= r15031964;
        double r15031966 = 4.16438922228;
        double r15031967 = y;
        double r15031968 = r15031967 / r15031963;
        double r15031969 = r15031968 / r15031963;
        double r15031970 = fma(r15031963, r15031966, r15031969);
        double r15031971 = 110.11392429848108;
        double r15031972 = r15031970 - r15031971;
        double r15031973 = 1498395052788.919;
        bool r15031974 = r15031963 <= r15031973;
        double r15031975 = 78.6994924154;
        double r15031976 = fma(r15031963, r15031966, r15031975);
        double r15031977 = 137.519416416;
        double r15031978 = fma(r15031963, r15031976, r15031977);
        double r15031979 = fma(r15031963, r15031978, r15031967);
        double r15031980 = z;
        double r15031981 = fma(r15031963, r15031979, r15031980);
        double r15031982 = 43.3400022514;
        double r15031983 = r15031963 + r15031982;
        double r15031984 = 263.505074721;
        double r15031985 = fma(r15031983, r15031963, r15031984);
        double r15031986 = 313.399215894;
        double r15031987 = fma(r15031985, r15031963, r15031986);
        double r15031988 = 47.066876606;
        double r15031989 = fma(r15031987, r15031963, r15031988);
        double r15031990 = r15031963 * r15031963;
        double r15031991 = 2.0;
        double r15031992 = r15031991 * r15031991;
        double r15031993 = r15031990 - r15031992;
        double r15031994 = r15031989 / r15031993;
        double r15031995 = r15031963 + r15031991;
        double r15031996 = r15031994 * r15031995;
        double r15031997 = r15031981 / r15031996;
        double r15031998 = r15031967 / r15031990;
        double r15031999 = 110.1139242984811;
        double r15032000 = r15031998 - r15031999;
        double r15032001 = fma(r15031966, r15031963, r15032000);
        double r15032002 = r15031974 ? r15031997 : r15032001;
        double r15032003 = r15031965 ? r15031972 : r15032002;
        return r15032003;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.6
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -9.097538303619485e+29

    1. Initial program 58.6

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

    2. Simplified54.3

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

    4. Applied flip--54.3

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

    5. Applied associate-/r/54.3

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

    6. Taylor expanded around inf 1.4

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810806547102401964366436005}\]

    7. Simplified1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{\frac{y}{x}}{x}\right) - 110.1139242984810806547102401964366436005}\]

    if -9.097538303619485e+29 < x < 1498395052788.919

    1. Initial program 0.3

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

    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 flip--0.2

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

    5. Applied associate-/r/0.2

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

    if 1498395052788.919 < x

    1. Initial program 54.5

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

    2. Simplified50.1

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

    3. Taylor expanded around inf 2.3

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

    4. Simplified2.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -909753830361948524383850987520:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{\frac{y}{x}}{x}\right) - 110.1139242984810806547102401964366436005\\ \mathbf{elif}\;x \le 1498395052788.9189453125:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x \cdot x - 2 \cdot 2} \cdot \left(x + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x} - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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)))