Average Error: 29.5 → 5.0
Time: 6.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.6077243051632657 \cdot 10^{52} \lor \neg \left(z \le 5545609058328270340000\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}\\ \end{array}\]
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 -3.6077243051632657 \cdot 10^{52} \lor \neg \left(z \le 5545609058328270340000\right):\\
\;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\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 \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}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r391642 = x;
        double r391643 = y;
        double r391644 = z;
        double r391645 = 3.13060547623;
        double r391646 = r391644 * r391645;
        double r391647 = 11.1667541262;
        double r391648 = r391646 + r391647;
        double r391649 = r391648 * r391644;
        double r391650 = t;
        double r391651 = r391649 + r391650;
        double r391652 = r391651 * r391644;
        double r391653 = a;
        double r391654 = r391652 + r391653;
        double r391655 = r391654 * r391644;
        double r391656 = b;
        double r391657 = r391655 + r391656;
        double r391658 = r391643 * r391657;
        double r391659 = 15.234687407;
        double r391660 = r391644 + r391659;
        double r391661 = r391660 * r391644;
        double r391662 = 31.4690115749;
        double r391663 = r391661 + r391662;
        double r391664 = r391663 * r391644;
        double r391665 = 11.9400905721;
        double r391666 = r391664 + r391665;
        double r391667 = r391666 * r391644;
        double r391668 = 0.607771387771;
        double r391669 = r391667 + r391668;
        double r391670 = r391658 / r391669;
        double r391671 = r391642 + r391670;
        return r391671;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r391672 = z;
        double r391673 = -3.6077243051632657e+52;
        bool r391674 = r391672 <= r391673;
        double r391675 = 5.54560905832827e+21;
        bool r391676 = r391672 <= r391675;
        double r391677 = !r391676;
        bool r391678 = r391674 || r391677;
        double r391679 = x;
        double r391680 = 3.13060547623;
        double r391681 = y;
        double r391682 = r391680 * r391681;
        double r391683 = t;
        double r391684 = r391683 * r391681;
        double r391685 = 2.0;
        double r391686 = pow(r391672, r391685);
        double r391687 = r391684 / r391686;
        double r391688 = r391682 + r391687;
        double r391689 = 36.527041698806414;
        double r391690 = r391681 / r391672;
        double r391691 = r391689 * r391690;
        double r391692 = r391688 - r391691;
        double r391693 = r391679 + r391692;
        double r391694 = 15.234687407;
        double r391695 = r391672 + r391694;
        double r391696 = r391695 * r391672;
        double r391697 = 31.4690115749;
        double r391698 = r391696 + r391697;
        double r391699 = r391698 * r391672;
        double r391700 = 11.9400905721;
        double r391701 = r391699 + r391700;
        double r391702 = r391701 * r391672;
        double r391703 = 0.607771387771;
        double r391704 = r391702 + r391703;
        double r391705 = sqrt(r391704);
        double r391706 = r391681 / r391705;
        double r391707 = r391672 * r391680;
        double r391708 = 11.1667541262;
        double r391709 = r391707 + r391708;
        double r391710 = r391709 * r391672;
        double r391711 = r391710 + r391683;
        double r391712 = r391711 * r391672;
        double r391713 = a;
        double r391714 = r391712 + r391713;
        double r391715 = r391714 * r391672;
        double r391716 = b;
        double r391717 = r391715 + r391716;
        double r391718 = r391717 / r391705;
        double r391719 = r391706 * r391718;
        double r391720 = r391679 + r391719;
        double r391721 = r391678 ? r391693 : r391720;
        return r391721;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target1.1
Herbie5.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.6077243051632657e+52 or 5.54560905832827e+21 < z

    1. Initial program 59.7

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

    2. Taylor expanded around inf 9.1

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

    if -3.6077243051632657e+52 < z < 5.54560905832827e+21

    1. Initial program 1.8

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

    3. Applied add-sqr-sqrt2.3

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

    4. Applied times-frac1.2

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.6077243051632657 \cdot 10^{52} \lor \neg \left(z \le 5545609058328270340000\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))