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

Average Error: 20.5 → 0.1

Time: 13.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -107162863495500986155270144 \lor \neg \left(z \le 67465969353685.640625\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r258895 = x;
        double r258896 = y;
        double r258897 = z;
        double r258898 = 0.0692910599291889;
        double r258899 = r258897 * r258898;
        double r258900 = 0.4917317610505968;
        double r258901 = r258899 + r258900;
        double r258902 = r258901 * r258897;
        double r258903 = 0.279195317918525;
        double r258904 = r258902 + r258903;
        double r258905 = r258896 * r258904;
        double r258906 = 6.012459259764103;
        double r258907 = r258897 + r258906;
        double r258908 = r258907 * r258897;
        double r258909 = 3.350343815022304;
        double r258910 = r258908 + r258909;
        double r258911 = r258905 / r258910;
        double r258912 = r258895 + r258911;
        return r258912;
}

↓

double f(double x, double y, double z) {
        double r258913 = z;
        double r258914 = -1.0716286349550099e+26;
        bool r258915 = r258913 <= r258914;
        double r258916 = 67465969353685.64;
        bool r258917 = r258913 <= r258916;
        double r258918 = !r258917;
        bool r258919 = r258915 || r258918;
        double r258920 = x;
        double r258921 = y;
        double r258922 = 0.07512208616047561;
        double r258923 = 1.0;
        double r258924 = r258923 / r258913;
        double r258925 = r258922 * r258924;
        double r258926 = 0.0692910599291889;
        double r258927 = r258925 + r258926;
        double r258928 = 0.40462203869992125;
        double r258929 = 2.0;
        double r258930 = pow(r258913, r258929);
        double r258931 = r258923 / r258930;
        double r258932 = r258928 * r258931;
        double r258933 = r258927 - r258932;
        double r258934 = r258921 * r258933;
        double r258935 = r258920 + r258934;
        double r258936 = r258913 * r258926;
        double r258937 = 0.4917317610505968;
        double r258938 = r258936 + r258937;
        double r258939 = r258938 * r258913;
        double r258940 = 0.279195317918525;
        double r258941 = r258939 + r258940;
        double r258942 = 6.012459259764103;
        double r258943 = r258913 + r258942;
        double r258944 = r258943 * r258913;
        double r258945 = 3.350343815022304;
        double r258946 = r258944 + r258945;
        double r258947 = r258941 / r258946;
        double r258948 = r258921 * r258947;
        double r258949 = r258920 + r258948;
        double r258950 = r258919 ? r258935 : r258949;
        return r258950;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.5
Target	0.1
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 < -1.0716286349550099e+26 or 67465969353685.64 < z

   1. Initial program 43.6

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

   3. Applied *-un-lft-identity 43.6

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

   4. Applied times-frac 35.5

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

   5. Simplified 35.5

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

   6. Taylor expanded around inf 0.0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)}\]

   if -1.0716286349550099e+26 < z < 67465969353685.64

   1. Initial program 0.3

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

   3. Applied *-un-lft-identity 0.3

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -107162863495500986155270144 \lor \neg \left(z \le 67465969353685.640625\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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))))