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

Average Error: 19.9 → 0.1

Time: 24.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -40697775657662409605120:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \mathbf{elif}\;z \le 212668326.6286086738109588623046875:\\ \;\;\;\;\left(y \cdot \frac{1}{\sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} \cdot \sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} + x\\ \mathbf{else}:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r17511607 = x;
        double r17511608 = y;
        double r17511609 = z;
        double r17511610 = 0.0692910599291889;
        double r17511611 = r17511609 * r17511610;
        double r17511612 = 0.4917317610505968;
        double r17511613 = r17511611 + r17511612;
        double r17511614 = r17511613 * r17511609;
        double r17511615 = 0.279195317918525;
        double r17511616 = r17511614 + r17511615;
        double r17511617 = r17511608 * r17511616;
        double r17511618 = 6.012459259764103;
        double r17511619 = r17511609 + r17511618;
        double r17511620 = r17511619 * r17511609;
        double r17511621 = 3.350343815022304;
        double r17511622 = r17511620 + r17511621;
        double r17511623 = r17511617 / r17511622;
        double r17511624 = r17511607 + r17511623;
        return r17511624;
}

↓

double f(double x, double y, double z) {
        double r17511625 = z;
        double r17511626 = -4.069777565766241e+22;
        bool r17511627 = r17511625 <= r17511626;
        double r17511628 = 0.0692910599291889;
        double r17511629 = y;
        double r17511630 = r17511628 * r17511629;
        double r17511631 = r17511629 / r17511625;
        double r17511632 = 0.07512208616047561;
        double r17511633 = r17511631 * r17511632;
        double r17511634 = r17511630 + r17511633;
        double r17511635 = x;
        double r17511636 = r17511634 + r17511635;
        double r17511637 = 212668326.62860867;
        bool r17511638 = r17511625 <= r17511637;
        double r17511639 = 1.0;
        double r17511640 = 6.012459259764103;
        double r17511641 = r17511640 + r17511625;
        double r17511642 = r17511625 * r17511641;
        double r17511643 = 3.350343815022304;
        double r17511644 = r17511642 + r17511643;
        double r17511645 = cbrt(r17511644);
        double r17511646 = sqrt(r17511645);
        double r17511647 = r17511645 * r17511645;
        double r17511648 = sqrt(r17511647);
        double r17511649 = r17511646 * r17511648;
        double r17511650 = r17511639 / r17511649;
        double r17511651 = r17511629 * r17511650;
        double r17511652 = r17511625 * r17511628;
        double r17511653 = 0.4917317610505968;
        double r17511654 = r17511652 + r17511653;
        double r17511655 = r17511654 * r17511625;
        double r17511656 = 0.279195317918525;
        double r17511657 = r17511655 + r17511656;
        double r17511658 = sqrt(r17511644);
        double r17511659 = r17511657 / r17511658;
        double r17511660 = r17511651 * r17511659;
        double r17511661 = r17511660 + r17511635;
        double r17511662 = r17511638 ? r17511661 : r17511636;
        double r17511663 = r17511627 ? r17511636 : r17511662;
        return r17511663;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.9
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 < -4.069777565766241e+22 or 212668326.62860867 < z

   1. Initial program 41.8

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

   2. Taylor expanded around inf 0.0

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

   if -4.069777565766241e+22 < z < 212668326.62860867

   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 *-un-lft-identity 0.2

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

   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

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

   7. Applied add-sqr-sqrt 0.4

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

   8. Applied *-un-lft-identity 0.4

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

   9. Applied times-frac 0.2

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

   10. Applied associate-*r* 0.2

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

   12. Applied add-cube-cbrt 0.2

       \[\leadsto x + \left(y \cdot \frac{1}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}\right) \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]

   13. Applied sqrt-prod 0.2

       \[\leadsto x + \left(y \cdot \frac{1}{\color{blue}{\sqrt{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}} \cdot \sqrt{\sqrt[3]{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{\left(z + 6.012459259764103336465268512256443500519\right) \cdot z + 3.350343815022303939343828460550867021084}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -40697775657662409605120:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \mathbf{elif}\;z \le 212668326.6286086738109588623046875:\\ \;\;\;\;\left(y \cdot \frac{1}{\sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} \cdot \sqrt{\sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084} \cdot \sqrt[3]{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}}}\right) \cdot \frac{\left(z \cdot 0.06929105992918889456166908757950295694172 + 0.4917317610505967939715787906607147306204\right) \cdot z + 0.2791953179185249767080279070796677842736}{\sqrt{z \cdot \left(6.012459259764103336465268512256443500519 + z\right) + 3.350343815022303939343828460550867021084}} + x\\ \mathbf{else}:\\ \;\;\;\;\left(0.06929105992918889456166908757950295694172 \cdot y + \frac{y}{z} \cdot 0.07512208616047560960637952121032867580652\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))