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

Average Error: 20.2 → 0.5

Time: 13.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.72325798806199381989684956555032586892 \cdot 10^{47} \lor \neg \left(z \le 1.879850085979871535558211275291510018991 \cdot 10^{-14}\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r310768 = x;
        double r310769 = y;
        double r310770 = z;
        double r310771 = 0.0692910599291889;
        double r310772 = r310770 * r310771;
        double r310773 = 0.4917317610505968;
        double r310774 = r310772 + r310773;
        double r310775 = r310774 * r310770;
        double r310776 = 0.279195317918525;
        double r310777 = r310775 + r310776;
        double r310778 = r310769 * r310777;
        double r310779 = 6.012459259764103;
        double r310780 = r310770 + r310779;
        double r310781 = r310780 * r310770;
        double r310782 = 3.350343815022304;
        double r310783 = r310781 + r310782;
        double r310784 = r310778 / r310783;
        double r310785 = r310768 + r310784;
        return r310785;
}

↓

double f(double x, double y, double z) {
        double r310786 = z;
        double r310787 = -1.7232579880619938e+47;
        bool r310788 = r310786 <= r310787;
        double r310789 = 1.8798500859798715e-14;
        bool r310790 = r310786 <= r310789;
        double r310791 = !r310790;
        bool r310792 = r310788 || r310791;
        double r310793 = x;
        double r310794 = y;
        double r310795 = 0.07512208616047561;
        double r310796 = 1.0;
        double r310797 = r310796 / r310786;
        double r310798 = r310795 * r310797;
        double r310799 = 0.0692910599291889;
        double r310800 = r310798 + r310799;
        double r310801 = 0.40462203869992125;
        double r310802 = 2.0;
        double r310803 = pow(r310786, r310802);
        double r310804 = r310796 / r310803;
        double r310805 = r310801 * r310804;
        double r310806 = r310800 - r310805;
        double r310807 = r310794 * r310806;
        double r310808 = r310793 + r310807;
        double r310809 = r310786 * r310799;
        double r310810 = 0.4917317610505968;
        double r310811 = r310809 + r310810;
        double r310812 = r310811 * r310786;
        double r310813 = 0.279195317918525;
        double r310814 = r310812 + r310813;
        double r310815 = 6.012459259764103;
        double r310816 = r310786 + r310815;
        double r310817 = r310816 * r310786;
        double r310818 = 3.350343815022304;
        double r310819 = r310817 + r310818;
        double r310820 = r310814 / r310819;
        double r310821 = r310794 * r310820;
        double r310822 = r310793 + r310821;
        double r310823 = r310792 ? r310808 : r310822;
        return r310823;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.2
Target	0.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if z < -1.7232579880619938e+47 or 1.8798500859798715e-14 < z

   1. Initial program 42.4

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

   3. Applied *-un-lft-identity 42.4

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

   4. Applied times-frac 34.1

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

   5. Simplified 34.1

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

   6. Taylor expanded around inf 0.9

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

   if -1.7232579880619938e+47 < z < 1.8798500859798715e-14

   1. Initial program 0.5

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

   3. Applied *-un-lft-identity 0.5

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

   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 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.72325798806199381989684956555032586892 \cdot 10^{47} \lor \neg \left(z \le 1.879850085979871535558211275291510018991 \cdot 10^{-14}\right):\\ \;\;\;\;x + y \cdot \left(\left(0.07512208616047560960637952121032867580652 \cdot \frac{1}{z} + 0.06929105992918889456166908757950295694172\right) - 0.4046220386999212492717958866705885156989 \cdot \frac{1}{{z}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))))