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

Average Error: 25.5 → 0.5

Time: 23.8s

Precision: 64

Math

\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.3070054425937903 \cdot 10^{+67}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \le 9.776865012149899 \cdot 10^{+58}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + \left(43.3400022514 \cdot \left(x \cdot x\right) + \left(x \cdot x + 263.505074721\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{y}{x \cdot x}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r17382131 = x;
        double r17382132 = 2.0;
        double r17382133 = r17382131 - r17382132;
        double r17382134 = 4.16438922228;
        double r17382135 = r17382131 * r17382134;
        double r17382136 = 78.6994924154;
        double r17382137 = r17382135 + r17382136;
        double r17382138 = r17382137 * r17382131;
        double r17382139 = 137.519416416;
        double r17382140 = r17382138 + r17382139;
        double r17382141 = r17382140 * r17382131;
        double r17382142 = y;
        double r17382143 = r17382141 + r17382142;
        double r17382144 = r17382143 * r17382131;
        double r17382145 = z;
        double r17382146 = r17382144 + r17382145;
        double r17382147 = r17382133 * r17382146;
        double r17382148 = 43.3400022514;
        double r17382149 = r17382131 + r17382148;
        double r17382150 = r17382149 * r17382131;
        double r17382151 = 263.505074721;
        double r17382152 = r17382150 + r17382151;
        double r17382153 = r17382152 * r17382131;
        double r17382154 = 313.399215894;
        double r17382155 = r17382153 + r17382154;
        double r17382156 = r17382155 * r17382131;
        double r17382157 = 47.066876606;
        double r17382158 = r17382156 + r17382157;
        double r17382159 = r17382147 / r17382158;
        return r17382159;
}

↓

double f(double x, double y, double z) {
        double r17382160 = x;
        double r17382161 = -4.3070054425937903e+67;
        bool r17382162 = r17382160 <= r17382161;
        double r17382163 = 4.16438922228;
        double r17382164 = r17382163 * r17382160;
        double r17382165 = 110.1139242984811;
        double r17382166 = r17382164 - r17382165;
        double r17382167 = y;
        double r17382168 = r17382160 * r17382160;
        double r17382169 = r17382167 / r17382168;
        double r17382170 = r17382166 + r17382169;
        double r17382171 = 9.776865012149899e+58;
        bool r17382172 = r17382160 <= r17382171;
        double r17382173 = 2.0;
        double r17382174 = r17382160 - r17382173;
        double r17382175 = z;
        double r17382176 = 137.519416416;
        double r17382177 = 78.6994924154;
        double r17382178 = r17382164 + r17382177;
        double r17382179 = r17382160 * r17382178;
        double r17382180 = r17382176 + r17382179;
        double r17382181 = r17382160 * r17382180;
        double r17382182 = r17382167 + r17382181;
        double r17382183 = r17382160 * r17382182;
        double r17382184 = r17382175 + r17382183;
        double r17382185 = 47.066876606;
        double r17382186 = 313.399215894;
        double r17382187 = 43.3400022514;
        double r17382188 = r17382187 * r17382168;
        double r17382189 = 263.505074721;
        double r17382190 = r17382168 + r17382189;
        double r17382191 = r17382190 * r17382160;
        double r17382192 = r17382188 + r17382191;
        double r17382193 = r17382186 + r17382192;
        double r17382194 = r17382160 * r17382193;
        double r17382195 = r17382185 + r17382194;
        double r17382196 = r17382184 / r17382195;
        double r17382197 = r17382174 * r17382196;
        double r17382198 = r17382172 ? r17382197 : r17382170;
        double r17382199 = r17382162 ? r17382170 : r17382198;
        return r17382199;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	25.5
Target	0.5
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -4.3070054425937903e+67 or 9.776865012149899e+58 < x

   1. Initial program 61.7

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

   2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922228 \cdot x\right) - 110.1139242984811}\]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{y}{x \cdot x}}\]

   if -4.3070054425937903e+67 < x < 9.776865012149899e+58

   1. Initial program 2.1

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 2.1

      \[\leadsto \frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606\right)}}\]

   4. Applied times-frac 0.7

      \[\leadsto \color{blue}{\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}\]

   5. Simplified 0.7

      \[\leadsto \color{blue}{\left(x - 2.0\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

   6. Taylor expanded around 0 0.7

      \[\leadsto \left(x - 2.0\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\left(263.505074721 \cdot x + \left({x}^{3} + 43.3400022514 \cdot {x}^{2}\right)\right)} + 313.399215894\right) \cdot x + 47.066876606}\]

   7. Simplified 0.7

      \[\leadsto \left(x - 2.0\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(263.505074721 + x \cdot x\right)\right)} + 313.399215894\right) \cdot x + 47.066876606}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.3070054425937903 \cdot 10^{+67}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \le 9.776865012149899 \cdot 10^{+58}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + \left(43.3400022514 \cdot \left(x \cdot x\right) + \left(x \cdot x + 263.505074721\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{y}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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) (/ (+ (* (+ (* (+ (* (+ (* 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)))