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

Average Error: 20.6 → 0.2

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -335377622387411131416765624483840:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{elif}\;z \le 696494.022742792847566306591033935546875:\\ \;\;\;\;x + \frac{y \cdot 0.2791953179185249767080279070796677842736 + y \cdot \left(\left(z \cdot z\right) \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204 \cdot z\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{z \cdot z}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r27729726 = x;
        double r27729727 = y;
        double r27729728 = z;
        double r27729729 = 0.0692910599291889;
        double r27729730 = r27729728 * r27729729;
        double r27729731 = 0.4917317610505968;
        double r27729732 = r27729730 + r27729731;
        double r27729733 = r27729732 * r27729728;
        double r27729734 = 0.279195317918525;
        double r27729735 = r27729733 + r27729734;
        double r27729736 = r27729727 * r27729735;
        double r27729737 = 6.012459259764103;
        double r27729738 = r27729728 + r27729737;
        double r27729739 = r27729738 * r27729728;
        double r27729740 = 3.350343815022304;
        double r27729741 = r27729739 + r27729740;
        double r27729742 = r27729736 / r27729741;
        double r27729743 = r27729726 + r27729742;
        return r27729743;
}

↓

double f(double x, double y, double z) {
        double r27729744 = z;
        double r27729745 = -3.353776223874111e+32;
        bool r27729746 = r27729744 <= r27729745;
        double r27729747 = x;
        double r27729748 = 0.0692910599291889;
        double r27729749 = y;
        double r27729750 = r27729748 * r27729749;
        double r27729751 = 0.07512208616047561;
        double r27729752 = r27729749 / r27729744;
        double r27729753 = r27729751 * r27729752;
        double r27729754 = 0.40462203869992125;
        double r27729755 = r27729744 * r27729744;
        double r27729756 = r27729749 / r27729755;
        double r27729757 = r27729754 * r27729756;
        double r27729758 = r27729753 - r27729757;
        double r27729759 = r27729750 + r27729758;
        double r27729760 = r27729747 + r27729759;
        double r27729761 = 696494.0227427928;
        bool r27729762 = r27729744 <= r27729761;
        double r27729763 = 0.279195317918525;
        double r27729764 = r27729749 * r27729763;
        double r27729765 = r27729755 * r27729748;
        double r27729766 = 0.4917317610505968;
        double r27729767 = r27729766 * r27729744;
        double r27729768 = r27729765 + r27729767;
        double r27729769 = r27729749 * r27729768;
        double r27729770 = r27729764 + r27729769;
        double r27729771 = 6.012459259764103;
        double r27729772 = r27729744 + r27729771;
        double r27729773 = r27729772 * r27729744;
        double r27729774 = 3.350343815022304;
        double r27729775 = r27729773 + r27729774;
        double r27729776 = r27729770 / r27729775;
        double r27729777 = r27729747 + r27729776;
        double r27729778 = r27729762 ? r27729777 : r27729760;
        double r27729779 = r27729746 ? r27729760 : r27729778;
        return r27729779;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.6
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if z < -3.353776223874111e+32 or 696494.0227427928 < z

   1. Initial program 43.1

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

   3. Applied add-sqr-sqrt 43.1

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

   4. Applied times-frac 34.5

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

   5. Taylor expanded around inf 0.0

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

   6. Simplified 0.0

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

   if -3.353776223874111e+32 < z < 696494.0227427928

   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. Taylor expanded around 0 0.3

      \[\leadsto x + \frac{\color{blue}{0.4917317610505967939715787906607147306204 \cdot \left(z \cdot y\right) + \left(0.06929105992918889456166908757950295694172 \cdot \left({z}^{2} \cdot y\right) + 0.2791953179185249767080279070796677842736 \cdot y\right)}}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\]

   3. Simplified 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -335377622387411131416765624483840:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{elif}\;z \le 696494.022742792847566306591033935546875:\\ \;\;\;\;x + \frac{y \cdot 0.2791953179185249767080279070796677842736 + y \cdot \left(\left(z \cdot z\right) \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204 \cdot z\right)}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.06929105992918889456166908757950295694172 \cdot y + \left(0.07512208616047560960637952121032867580652 \cdot \frac{y}{z} - 0.4046220386999212492717958866705885156989 \cdot \frac{y}{z \cdot z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))