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

Average Error: 26.6 → 1.0

Time: 7.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.63504100427729212 \cdot 10^{34}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \le 1.6132622111277306 \cdot 10^{68}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r472102 = x;
        double r472103 = 2.0;
        double r472104 = r472102 - r472103;
        double r472105 = 4.16438922228;
        double r472106 = r472102 * r472105;
        double r472107 = 78.6994924154;
        double r472108 = r472106 + r472107;
        double r472109 = r472108 * r472102;
        double r472110 = 137.519416416;
        double r472111 = r472109 + r472110;
        double r472112 = r472111 * r472102;
        double r472113 = y;
        double r472114 = r472112 + r472113;
        double r472115 = r472114 * r472102;
        double r472116 = z;
        double r472117 = r472115 + r472116;
        double r472118 = r472104 * r472117;
        double r472119 = 43.3400022514;
        double r472120 = r472102 + r472119;
        double r472121 = r472120 * r472102;
        double r472122 = 263.505074721;
        double r472123 = r472121 + r472122;
        double r472124 = r472123 * r472102;
        double r472125 = 313.399215894;
        double r472126 = r472124 + r472125;
        double r472127 = r472126 * r472102;
        double r472128 = 47.066876606;
        double r472129 = r472127 + r472128;
        double r472130 = r472118 / r472129;
        return r472130;
}

↓

double f(double x, double y, double z) {
        double r472131 = x;
        double r472132 = -2.635041004277292e+34;
        bool r472133 = r472131 <= r472132;
        double r472134 = 2.0;
        double r472135 = r472131 - r472134;
        double r472136 = y;
        double r472137 = 3.0;
        double r472138 = pow(r472131, r472137);
        double r472139 = r472136 / r472138;
        double r472140 = 4.16438922228;
        double r472141 = r472139 + r472140;
        double r472142 = 101.7851458539211;
        double r472143 = 1.0;
        double r472144 = r472143 / r472131;
        double r472145 = r472142 * r472144;
        double r472146 = r472141 - r472145;
        double r472147 = r472135 * r472146;
        double r472148 = 1.6132622111277306e+68;
        bool r472149 = r472131 <= r472148;
        double r472150 = 43.3400022514;
        double r472151 = r472131 + r472150;
        double r472152 = r472151 * r472131;
        double r472153 = 263.505074721;
        double r472154 = r472152 + r472153;
        double r472155 = r472154 * r472131;
        double r472156 = 313.399215894;
        double r472157 = r472155 + r472156;
        double r472158 = r472157 * r472131;
        double r472159 = 47.066876606;
        double r472160 = r472158 + r472159;
        double r472161 = sqrt(r472160);
        double r472162 = r472143 / r472161;
        double r472163 = r472131 * r472140;
        double r472164 = 78.6994924154;
        double r472165 = r472163 + r472164;
        double r472166 = r472165 * r472131;
        double r472167 = 137.519416416;
        double r472168 = r472166 + r472167;
        double r472169 = r472168 * r472131;
        double r472170 = r472169 + r472136;
        double r472171 = r472170 * r472131;
        double r472172 = z;
        double r472173 = r472171 + r472172;
        double r472174 = r472173 / r472161;
        double r472175 = r472162 * r472174;
        double r472176 = r472135 * r472175;
        double r472177 = 2.0;
        double r472178 = pow(r472131, r472177);
        double r472179 = r472136 / r472178;
        double r472180 = r472140 * r472131;
        double r472181 = r472179 + r472180;
        double r472182 = 110.1139242984811;
        double r472183 = r472181 - r472182;
        double r472184 = r472149 ? r472176 : r472183;
        double r472185 = r472133 ? r472147 : r472184;
        return r472185;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.6
Target	0.5
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if x < -2.635041004277292e+34

   1. Initial program 59.1

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

   3. Applied *-un-lft-identity 59.1

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

   4. Applied times-frac 54.8

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

   5. Simplified 54.8

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

   6. Taylor expanded around inf 1.6

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)}\]

   if -2.635041004277292e+34 < x < 1.6132622111277306e+68

   1. Initial program 2.0

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

   3. Applied *-un-lft-identity 2.0

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

   4. Applied times-frac 0.6

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

   5. Simplified 0.6

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

   7. Applied add-sqr-sqrt 0.8

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

   8. Applied *-un-lft-identity 0.8

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

   9. Applied times-frac 1.0

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

   if 1.6132622111277306e+68 < x

   1. Initial program 64.0

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

   2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.63504100427729212 \cdot 10^{34}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \le 1.6132622111277306 \cdot 10^{68}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))