Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 26.8 → 0.6

Time: 14.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.37322170457412272 \cdot 10^{60} \lor \neg \left(x \le 4.54262428893722098 \cdot 10^{64}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r337959 = x;
        double r337960 = 2.0;
        double r337961 = r337959 - r337960;
        double r337962 = 4.16438922228;
        double r337963 = r337959 * r337962;
        double r337964 = 78.6994924154;
        double r337965 = r337963 + r337964;
        double r337966 = r337965 * r337959;
        double r337967 = 137.519416416;
        double r337968 = r337966 + r337967;
        double r337969 = r337968 * r337959;
        double r337970 = y;
        double r337971 = r337969 + r337970;
        double r337972 = r337971 * r337959;
        double r337973 = z;
        double r337974 = r337972 + r337973;
        double r337975 = r337961 * r337974;
        double r337976 = 43.3400022514;
        double r337977 = r337959 + r337976;
        double r337978 = r337977 * r337959;
        double r337979 = 263.505074721;
        double r337980 = r337978 + r337979;
        double r337981 = r337980 * r337959;
        double r337982 = 313.399215894;
        double r337983 = r337981 + r337982;
        double r337984 = r337983 * r337959;
        double r337985 = 47.066876606;
        double r337986 = r337984 + r337985;
        double r337987 = r337975 / r337986;
        return r337987;
}

↓

double f(double x, double y, double z) {
        double r337988 = x;
        double r337989 = -4.373221704574123e+60;
        bool r337990 = r337988 <= r337989;
        double r337991 = 4.542624288937221e+64;
        bool r337992 = r337988 <= r337991;
        double r337993 = !r337992;
        bool r337994 = r337990 || r337993;
        double r337995 = y;
        double r337996 = 2.0;
        double r337997 = pow(r337988, r337996);
        double r337998 = r337995 / r337997;
        double r337999 = 4.16438922228;
        double r338000 = r337999 * r337988;
        double r338001 = r337998 + r338000;
        double r338002 = 110.1139242984811;
        double r338003 = r338001 - r338002;
        double r338004 = r337988 * r337999;
        double r338005 = 78.6994924154;
        double r338006 = r338004 + r338005;
        double r338007 = r338006 * r337988;
        double r338008 = 137.519416416;
        double r338009 = r338007 + r338008;
        double r338010 = r338009 * r337988;
        double r338011 = r338010 + r337995;
        double r338012 = r338011 * r337988;
        double r338013 = z;
        double r338014 = r338012 + r338013;
        double r338015 = 2.0;
        double r338016 = r337988 - r338015;
        double r338017 = 43.3400022514;
        double r338018 = r337988 + r338017;
        double r338019 = r338018 * r337988;
        double r338020 = 263.505074721;
        double r338021 = r338019 + r338020;
        double r338022 = r338021 * r337988;
        double r338023 = 313.399215894;
        double r338024 = r338022 + r338023;
        double r338025 = r338024 * r337988;
        double r338026 = 47.066876606;
        double r338027 = r338025 + r338026;
        double r338028 = r338016 / r338027;
        double r338029 = r338014 * r338028;
        double r338030 = r337994 ? r338003 : r338029;
        return r338030;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.8
Target	0.4
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -4.373221704574123e+60 or 4.542624288937221e+64 < x

   1. Initial program 63.9

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied associate-/l* 61.0

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 61.0

      \[\leadsto \frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\color{blue}{1 \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}}}\]

   6. Applied *-un-lft-identity 61.0

      \[\leadsto \frac{x - 2}{\frac{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}{1 \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}}\]

   7. Applied times-frac 61.0

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{1}{1} \cdot \frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}}\]

   8. Applied *-un-lft-identity 61.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x - 2\right)}}{\frac{1}{1} \cdot \frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}\]

   9. Applied times-frac 61.0

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1}} \cdot \frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}}\]

   10. Simplified 61.0

       \[\leadsto \color{blue}{1} \cdot \frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}\]

   11. Simplified 61.0

       \[\leadsto 1 \cdot \color{blue}{\left(\left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\right)}\]

   12. Taylor expanded around inf 0.2

       \[\leadsto 1 \cdot \color{blue}{\left(\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\right)}\]

   if -4.373221704574123e+60 < x < 4.542624288937221e+64

   1. Initial program 1.9

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.8

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 0.8

      \[\leadsto \frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\color{blue}{1 \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}}}\]

   6. Applied *-un-lft-identity 0.8

      \[\leadsto \frac{x - 2}{\frac{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}{1 \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}}\]

   7. Applied times-frac 0.8

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{1}{1} \cdot \frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}}\]

   8. Applied *-un-lft-identity 0.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x - 2\right)}}{\frac{1}{1} \cdot \frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}\]

   9. Applied times-frac 0.8

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1}} \cdot \frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}}\]

   10. Simplified 0.8

       \[\leadsto \color{blue}{1} \cdot \frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}}\]

   11. Simplified 0.9

       \[\leadsto 1 \cdot \color{blue}{\left(\left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.37322170457412272 \cdot 10^{60} \lor \neg \left(x \le 4.54262428893722098 \cdot 10^{64}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))