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

Average Error: 20.6 → 0.2

Time: 1.1m

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r18175676 = x;
        double r18175677 = y;
        double r18175678 = z;
        double r18175679 = 0.0692910599291889;
        double r18175680 = r18175678 * r18175679;
        double r18175681 = 0.4917317610505968;
        double r18175682 = r18175680 + r18175681;
        double r18175683 = r18175682 * r18175678;
        double r18175684 = 0.279195317918525;
        double r18175685 = r18175683 + r18175684;
        double r18175686 = r18175677 * r18175685;
        double r18175687 = 6.012459259764103;
        double r18175688 = r18175678 + r18175687;
        double r18175689 = r18175688 * r18175678;
        double r18175690 = 3.350343815022304;
        double r18175691 = r18175689 + r18175690;
        double r18175692 = r18175686 / r18175691;
        double r18175693 = r18175676 + r18175692;
        return r18175693;
}

↓

double f(double x, double y, double z) {
        double r18175694 = z;
        double r18175695 = -3.353776223874111e+32;
        bool r18175696 = r18175694 <= r18175695;
        double r18175697 = 0.0692910599291889;
        double r18175698 = y;
        double r18175699 = r18175697 * r18175698;
        double r18175700 = 0.07512208616047561;
        double r18175701 = r18175694 / r18175698;
        double r18175702 = r18175700 / r18175701;
        double r18175703 = 0.40462203869992125;
        double r18175704 = r18175703 / r18175694;
        double r18175705 = r18175698 / r18175694;
        double r18175706 = r18175704 * r18175705;
        double r18175707 = r18175702 - r18175706;
        double r18175708 = r18175699 + r18175707;
        double r18175709 = x;
        double r18175710 = r18175708 + r18175709;
        double r18175711 = 696494.0227427928;
        bool r18175712 = r18175694 <= r18175711;
        double r18175713 = 0.4917317610505968;
        double r18175714 = r18175713 * r18175694;
        double r18175715 = r18175694 * r18175694;
        double r18175716 = r18175715 * r18175697;
        double r18175717 = r18175714 + r18175716;
        double r18175718 = r18175698 * r18175717;
        double r18175719 = 0.279195317918525;
        double r18175720 = r18175698 * r18175719;
        double r18175721 = r18175718 + r18175720;
        double r18175722 = 6.012459259764103;
        double r18175723 = r18175694 + r18175722;
        double r18175724 = r18175723 * r18175694;
        double r18175725 = 3.350343815022304;
        double r18175726 = r18175724 + r18175725;
        double r18175727 = r18175721 / r18175726;
        double r18175728 = r18175709 + r18175727;
        double r18175729 = r18175712 ? r18175728 : r18175710;
        double r18175730 = r18175696 ? r18175710 : r18175729;
        return r18175730;
}

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

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

   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

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

4. Final simplification 0.2

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

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))))