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

Average Error: 27.5 → 0.5

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.67035503993276071 \cdot 10^{43} \lor \neg \left(x \le 3.7248869126440915 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r494070 = x;
        double r494071 = 2.0;
        double r494072 = r494070 - r494071;
        double r494073 = 4.16438922228;
        double r494074 = r494070 * r494073;
        double r494075 = 78.6994924154;
        double r494076 = r494074 + r494075;
        double r494077 = r494076 * r494070;
        double r494078 = 137.519416416;
        double r494079 = r494077 + r494078;
        double r494080 = r494079 * r494070;
        double r494081 = y;
        double r494082 = r494080 + r494081;
        double r494083 = r494082 * r494070;
        double r494084 = z;
        double r494085 = r494083 + r494084;
        double r494086 = r494072 * r494085;
        double r494087 = 43.3400022514;
        double r494088 = r494070 + r494087;
        double r494089 = r494088 * r494070;
        double r494090 = 263.505074721;
        double r494091 = r494089 + r494090;
        double r494092 = r494091 * r494070;
        double r494093 = 313.399215894;
        double r494094 = r494092 + r494093;
        double r494095 = r494094 * r494070;
        double r494096 = 47.066876606;
        double r494097 = r494095 + r494096;
        double r494098 = r494086 / r494097;
        return r494098;
}

↓

double f(double x, double y, double z) {
        double r494099 = x;
        double r494100 = -3.6703550399327607e+43;
        bool r494101 = r494099 <= r494100;
        double r494102 = 3.7248869126440915e+43;
        bool r494103 = r494099 <= r494102;
        double r494104 = !r494103;
        bool r494105 = r494101 || r494104;
        double r494106 = y;
        double r494107 = 2.0;
        double r494108 = pow(r494099, r494107);
        double r494109 = r494106 / r494108;
        double r494110 = 4.16438922228;
        double r494111 = r494110 * r494099;
        double r494112 = r494109 + r494111;
        double r494113 = 110.1139242984811;
        double r494114 = r494112 - r494113;
        double r494115 = 2.0;
        double r494116 = r494099 - r494115;
        double r494117 = r494099 * r494110;
        double r494118 = 78.6994924154;
        double r494119 = r494117 + r494118;
        double r494120 = r494119 * r494099;
        double r494121 = 3.0;
        double r494122 = pow(r494120, r494121);
        double r494123 = 137.519416416;
        double r494124 = pow(r494123, r494121);
        double r494125 = r494122 + r494124;
        double r494126 = r494125 * r494099;
        double r494127 = r494120 * r494120;
        double r494128 = r494123 * r494123;
        double r494129 = r494120 * r494123;
        double r494130 = r494128 - r494129;
        double r494131 = r494127 + r494130;
        double r494132 = r494126 / r494131;
        double r494133 = r494132 + r494106;
        double r494134 = r494133 * r494099;
        double r494135 = z;
        double r494136 = r494134 + r494135;
        double r494137 = 43.3400022514;
        double r494138 = r494099 + r494137;
        double r494139 = r494138 * r494099;
        double r494140 = 263.505074721;
        double r494141 = r494139 + r494140;
        double r494142 = r494141 * r494099;
        double r494143 = 313.399215894;
        double r494144 = r494142 + r494143;
        double r494145 = r494144 * r494099;
        double r494146 = 47.066876606;
        double r494147 = r494145 + r494146;
        double r494148 = r494136 / r494147;
        double r494149 = r494116 * r494148;
        double r494150 = r494105 ? r494114 : r494149;
        return r494150;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.5
Target	0.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if x < -3.6703550399327607e+43 or 3.7248869126440915e+43 < x

   1. Initial program 61.0

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

   2. Taylor expanded around inf 0.6

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

   if -3.6703550399327607e+43 < x < 3.7248869126440915e+43

   1. Initial program 1.1

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

   3. Applied *-un-lft-identity 1.1

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

   4. Applied times-frac 0.4

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

   5. Simplified 0.4

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

   7. Applied flip3-+ 0.5

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\color{blue}{\frac{{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   8. Applied associate-*l/ 0.5

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.67035503993276071 \cdot 10^{43} \lor \neg \left(x \le 3.7248869126440915 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))