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

Average Error: 20.4 → 0.1

Time: 14.9s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1043089.88380932458676397800445556640625 \lor \neg \left(z \le 57436210746.768341064453125\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y \cdot \left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right)\right) \cdot z + 0.2791953179185249767080279070796677842736 \cdot y}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r302882 = x;
        double r302883 = y;
        double r302884 = z;
        double r302885 = 0.0692910599291889;
        double r302886 = r302884 * r302885;
        double r302887 = 0.4917317610505968;
        double r302888 = r302886 + r302887;
        double r302889 = r302888 * r302884;
        double r302890 = 0.279195317918525;
        double r302891 = r302889 + r302890;
        double r302892 = r302883 * r302891;
        double r302893 = 6.012459259764103;
        double r302894 = r302884 + r302893;
        double r302895 = r302894 * r302884;
        double r302896 = 3.350343815022304;
        double r302897 = r302895 + r302896;
        double r302898 = r302892 / r302897;
        double r302899 = r302882 + r302898;
        return r302899;
}

↓

double f(double x, double y, double z) {
        double r302900 = z;
        double r302901 = -1043089.8838093246;
        bool r302902 = r302900 <= r302901;
        double r302903 = 57436210746.76834;
        bool r302904 = r302900 <= r302903;
        double r302905 = !r302904;
        bool r302906 = r302902 || r302905;
        double r302907 = x;
        double r302908 = 0.0692910599291889;
        double r302909 = y;
        double r302910 = r302908 * r302909;
        double r302911 = r302909 / r302900;
        double r302912 = 0.07512208616047561;
        double r302913 = 0.40462203869992125;
        double r302914 = r302913 / r302900;
        double r302915 = r302912 - r302914;
        double r302916 = r302911 * r302915;
        double r302917 = r302910 + r302916;
        double r302918 = r302907 + r302917;
        double r302919 = r302900 * r302908;
        double r302920 = 0.4917317610505968;
        double r302921 = r302919 + r302920;
        double r302922 = r302909 * r302921;
        double r302923 = r302922 * r302900;
        double r302924 = 0.279195317918525;
        double r302925 = r302924 * r302909;
        double r302926 = r302923 + r302925;
        double r302927 = 6.012459259764103;
        double r302928 = r302900 + r302927;
        double r302929 = r302928 * r302900;
        double r302930 = 3.350343815022304;
        double r302931 = r302929 + r302930;
        double r302932 = r302926 / r302931;
        double r302933 = r302907 + r302932;
        double r302934 = r302906 ? r302918 : r302933;
        return r302934;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.4
Target	0.2
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;z \lt -8120153.6524566747248172760009765625:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z \lt 657611897278737678336:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047560960637952121032867580652}{z} + 0.06929105992918889456166908757950295694172\right) \cdot y - \left(\frac{0.4046220386999212492717958866705885156989 \cdot y}{z \cdot z} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1043089.8838093246 or 57436210746.76834 < z

   1. Initial program 41.9

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 41.9

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\color{blue}{1 \cdot \left(\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084\right)}}\]

   4. Applied times-frac 33.7

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]

   5. Simplified 33.7

      \[\leadsto x + \color{blue}{y} \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

   6. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} + 0.06929105992918889456166908757950295694172 \cdot y\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{{z}^{2}}\right)}\]

   7. Simplified 0.0

      \[\leadsto x + \color{blue}{\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)}\]

   if -1043089.8838093246 < z < 57436210746.76834

   1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
   2. Using strategy rm

   3. Applied distribute-lft-in 0.2

      \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z\right) + y \cdot 0.2791953179185249767080279070796677842736}}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

   4. Simplified 0.2

      \[\leadsto x + \frac{\color{blue}{\left(y \cdot \left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right)\right) \cdot z} + y \cdot 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

   5. Simplified 0.2

      \[\leadsto x + \frac{\left(y \cdot \left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right)\right) \cdot z + \color{blue}{0.2791953179185249767080279070796677842736 \cdot y}}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1043089.88380932458676397800445556640625 \lor \neg \left(z \le 57436210746.768341064453125\right):\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot \left(0.07512208616047560960637952121032867580652 - \frac{0.4046220386999212492717958866705885156989}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y \cdot \left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right)\right) \cdot z + 0.2791953179185249767080279070796677842736 \cdot y}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.6524566747) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x)) (if (< z 657611897278737680000) (+ x (* (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (/ 1 (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291888946) y) (- (/ (* 0.404622038699921249 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291888946) 0.49173176105059679) z) 0.279195317918524977)) (+ (* (+ z 6.0124592597641033) z) 3.35034381502230394))))