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

Average Error: 25.4 → 0.8

Time: 29.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.508874372053853 \cdot 10^{+28}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922228 - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 9.203477273753945 \cdot 10^{+42}:\\ \;\;\;\;\frac{\left(y + \left(137.519416416 + x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)\right) \cdot x\right) \cdot x + z}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \left(\frac{1}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \frac{x - 2.0}{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922228 - 110.1139242984811\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r21198090 = x;
        double r21198091 = 2.0;
        double r21198092 = r21198090 - r21198091;
        double r21198093 = 4.16438922228;
        double r21198094 = r21198090 * r21198093;
        double r21198095 = 78.6994924154;
        double r21198096 = r21198094 + r21198095;
        double r21198097 = r21198096 * r21198090;
        double r21198098 = 137.519416416;
        double r21198099 = r21198097 + r21198098;
        double r21198100 = r21198099 * r21198090;
        double r21198101 = y;
        double r21198102 = r21198100 + r21198101;
        double r21198103 = r21198102 * r21198090;
        double r21198104 = z;
        double r21198105 = r21198103 + r21198104;
        double r21198106 = r21198092 * r21198105;
        double r21198107 = 43.3400022514;
        double r21198108 = r21198090 + r21198107;
        double r21198109 = r21198108 * r21198090;
        double r21198110 = 263.505074721;
        double r21198111 = r21198109 + r21198110;
        double r21198112 = r21198111 * r21198090;
        double r21198113 = 313.399215894;
        double r21198114 = r21198112 + r21198113;
        double r21198115 = r21198114 * r21198090;
        double r21198116 = 47.066876606;
        double r21198117 = r21198115 + r21198116;
        double r21198118 = r21198106 / r21198117;
        return r21198118;
}

↓

double f(double x, double y, double z) {
        double r21198119 = x;
        double r21198120 = -2.508874372053853e+28;
        bool r21198121 = r21198119 <= r21198120;
        double r21198122 = y;
        double r21198123 = r21198119 * r21198119;
        double r21198124 = r21198122 / r21198123;
        double r21198125 = 4.16438922228;
        double r21198126 = r21198119 * r21198125;
        double r21198127 = 110.1139242984811;
        double r21198128 = r21198126 - r21198127;
        double r21198129 = r21198124 + r21198128;
        double r21198130 = 9.203477273753945e+42;
        bool r21198131 = r21198119 <= r21198130;
        double r21198132 = 137.519416416;
        double r21198133 = 78.6994924154;
        double r21198134 = r21198126 + r21198133;
        double r21198135 = r21198119 * r21198134;
        double r21198136 = r21198132 + r21198135;
        double r21198137 = r21198136 * r21198119;
        double r21198138 = r21198122 + r21198137;
        double r21198139 = r21198138 * r21198119;
        double r21198140 = z;
        double r21198141 = r21198139 + r21198140;
        double r21198142 = 47.066876606;
        double r21198143 = 43.3400022514;
        double r21198144 = r21198143 + r21198119;
        double r21198145 = r21198119 * r21198144;
        double r21198146 = 263.505074721;
        double r21198147 = r21198145 + r21198146;
        double r21198148 = r21198119 * r21198147;
        double r21198149 = 313.399215894;
        double r21198150 = r21198148 + r21198149;
        double r21198151 = r21198119 * r21198150;
        double r21198152 = r21198142 + r21198151;
        double r21198153 = sqrt(r21198152);
        double r21198154 = sqrt(r21198153);
        double r21198155 = r21198141 / r21198154;
        double r21198156 = 1.0;
        double r21198157 = r21198156 / r21198154;
        double r21198158 = 2.0;
        double r21198159 = r21198119 - r21198158;
        double r21198160 = r21198159 / r21198153;
        double r21198161 = r21198157 * r21198160;
        double r21198162 = r21198155 * r21198161;
        double r21198163 = r21198131 ? r21198162 : r21198129;
        double r21198164 = r21198121 ? r21198129 : r21198163;
        return r21198164;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	25.4
Target	0.4
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if x < -2.508874372053853e+28 or 9.203477273753945e+42 < x

   1. Initial program 57.5

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

   3. Applied add-sqr-sqrt 57.5

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

   4. Applied times-frac 53.4

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

   6. Applied add-sqr-sqrt 53.4

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

   7. Applied sqrt-prod 53.4

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

   8. Applied *-un-lft-identity 53.4

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

   9. Applied times-frac 53.4

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

   10. Applied associate-*r* 53.4

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

   11. Taylor expanded around inf 1.1

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

   12. Simplified 1.1

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

   if -2.508874372053853e+28 < x < 9.203477273753945e+42

   1. Initial program 0.6

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

   3. Applied add-sqr-sqrt 0.7

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

   4. Applied times-frac 0.7

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

   6. Applied add-sqr-sqrt 0.7

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

   7. Applied sqrt-prod 0.8

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

   8. Applied *-un-lft-identity 0.8

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

   9. Applied times-frac 0.7

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

   10. Applied associate-*r* 0.6

       \[\leadsto \color{blue}{\left(\frac{x - 2.0}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \cdot \frac{1}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\sqrt{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.508874372053853 \cdot 10^{+28}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922228 - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 9.203477273753945 \cdot 10^{+42}:\\ \;\;\;\;\frac{\left(y + \left(137.519416416 + x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)\right) \cdot x\right) \cdot x + z}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \left(\frac{1}{\sqrt{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}} \cdot \frac{x - 2.0}{\sqrt{47.066876606 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514 + x\right) + 263.505074721\right) + 313.399215894\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922228 - 110.1139242984811\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))