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

Average Error: 29.7 → 1.6

Time: 7.4s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -37920752987725400 \lor \neg \left(z \le 7.35027383630455016 \cdot 10^{56}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t}{\frac{{z}^{2}}{y}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \left(\frac{1}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r374679 = x;
        double r374680 = y;
        double r374681 = z;
        double r374682 = 3.13060547623;
        double r374683 = r374681 * r374682;
        double r374684 = 11.1667541262;
        double r374685 = r374683 + r374684;
        double r374686 = r374685 * r374681;
        double r374687 = t;
        double r374688 = r374686 + r374687;
        double r374689 = r374688 * r374681;
        double r374690 = a;
        double r374691 = r374689 + r374690;
        double r374692 = r374691 * r374681;
        double r374693 = b;
        double r374694 = r374692 + r374693;
        double r374695 = r374680 * r374694;
        double r374696 = 15.234687407;
        double r374697 = r374681 + r374696;
        double r374698 = r374697 * r374681;
        double r374699 = 31.4690115749;
        double r374700 = r374698 + r374699;
        double r374701 = r374700 * r374681;
        double r374702 = 11.9400905721;
        double r374703 = r374701 + r374702;
        double r374704 = r374703 * r374681;
        double r374705 = 0.607771387771;
        double r374706 = r374704 + r374705;
        double r374707 = r374695 / r374706;
        double r374708 = r374679 + r374707;
        return r374708;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r374709 = z;
        double r374710 = -3.79207529877254e+16;
        bool r374711 = r374709 <= r374710;
        double r374712 = 7.35027383630455e+56;
        bool r374713 = r374709 <= r374712;
        double r374714 = !r374713;
        bool r374715 = r374711 || r374714;
        double r374716 = x;
        double r374717 = 3.13060547623;
        double r374718 = y;
        double r374719 = r374717 * r374718;
        double r374720 = t;
        double r374721 = 2.0;
        double r374722 = pow(r374709, r374721);
        double r374723 = r374722 / r374718;
        double r374724 = r374720 / r374723;
        double r374725 = r374719 + r374724;
        double r374726 = 36.527041698806414;
        double r374727 = r374718 / r374709;
        double r374728 = r374726 * r374727;
        double r374729 = r374725 - r374728;
        double r374730 = r374716 + r374729;
        double r374731 = 15.234687407;
        double r374732 = r374709 + r374731;
        double r374733 = r374732 * r374709;
        double r374734 = 31.4690115749;
        double r374735 = r374733 + r374734;
        double r374736 = r374735 * r374709;
        double r374737 = 11.9400905721;
        double r374738 = r374736 + r374737;
        double r374739 = r374738 * r374709;
        double r374740 = 0.607771387771;
        double r374741 = r374739 + r374740;
        double r374742 = sqrt(r374741);
        double r374743 = r374718 / r374742;
        double r374744 = 1.0;
        double r374745 = sqrt(r374742);
        double r374746 = r374744 / r374745;
        double r374747 = r374709 * r374717;
        double r374748 = 11.1667541262;
        double r374749 = r374747 + r374748;
        double r374750 = r374749 * r374709;
        double r374751 = r374750 + r374720;
        double r374752 = r374751 * r374709;
        double r374753 = a;
        double r374754 = r374752 + r374753;
        double r374755 = r374754 * r374709;
        double r374756 = b;
        double r374757 = r374755 + r374756;
        double r374758 = r374757 / r374745;
        double r374759 = r374746 * r374758;
        double r374760 = r374743 * r374759;
        double r374761 = r374716 + r374760;
        double r374762 = r374715 ? r374730 : r374761;
        return r374762;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	29.7
Target	1.3
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -3.79207529877254e+16 or 7.35027383630455e+56 < z

   1. Initial program 59.5

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

   2. Taylor expanded around inf 8.5

      \[\leadsto x + \color{blue}{\left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]
   3. Using strategy rm

   4. Applied associate-/l* 2.1

      \[\leadsto x + \left(\left(3.13060547622999996 \cdot y + \color{blue}{\frac{t}{\frac{{z}^{2}}{y}}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\]

   if -3.79207529877254e+16 < z < 7.35027383630455e+56

   1. Initial program 1.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 2.3

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \cdot \sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\]

   4. Applied times-frac 1.1

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 1.1

      \[\leadsto x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt{\color{blue}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \cdot \sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}}\]

   7. Applied sqrt-prod 1.1

      \[\leadsto x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\color{blue}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}}\]

   8. Applied *-un-lft-identity 1.1

      \[\leadsto x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\color{blue}{1 \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\]

   9. Applied times-frac 1.0

      \[\leadsto x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -37920752987725400 \lor \neg \left(z \le 7.35027383630455016 \cdot 10^{56}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t}{\frac{{z}^{2}}{y}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \left(\frac{1}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))