Average Error: 26.6 → 1.2
Time: 26.0s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -52919176326804272705496416256:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right) + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 1498395052788.9189453125:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot \left(137.5194164160000127594685181975364685059 + \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right) \cdot x\right) + y\right) + z\right) \cdot \left(x \cdot x - 2 \cdot 2\right)}{\left(2 + x\right) \cdot \left(\left(\left(\left(43.3400022514000013984514225739985704422 + x\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}
↓
\begin{array}{l}
\mathbf{if}\;x \le -52919176326804272705496416256:\\
\;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right) + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 1498395052788.9189453125:\\
\;\;\;\;\frac{\left(x \cdot \left(x \cdot \left(137.5194164160000127594685181975364685059 + \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right) \cdot x\right) + y\right) + z\right) \cdot \left(x \cdot x - 2 \cdot 2\right)}{\left(2 + x\right) \cdot \left(\left(\left(\left(43.3400022514000013984514225739985704422 + x\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\

\end{array}
double f(double x, double y, double z) {
        double r18214615 = x;
        double r18214616 = 2.0;
        double r18214617 = r18214615 - r18214616;
        double r18214618 = 4.16438922228;
        double r18214619 = r18214615 * r18214618;
        double r18214620 = 78.6994924154;
        double r18214621 = r18214619 + r18214620;
        double r18214622 = r18214621 * r18214615;
        double r18214623 = 137.519416416;
        double r18214624 = r18214622 + r18214623;
        double r18214625 = r18214624 * r18214615;
        double r18214626 = y;
        double r18214627 = r18214625 + r18214626;
        double r18214628 = r18214627 * r18214615;
        double r18214629 = z;
        double r18214630 = r18214628 + r18214629;
        double r18214631 = r18214617 * r18214630;
        double r18214632 = 43.3400022514;
        double r18214633 = r18214615 + r18214632;
        double r18214634 = r18214633 * r18214615;
        double r18214635 = 263.505074721;
        double r18214636 = r18214634 + r18214635;
        double r18214637 = r18214636 * r18214615;
        double r18214638 = 313.399215894;
        double r18214639 = r18214637 + r18214638;
        double r18214640 = r18214639 * r18214615;
        double r18214641 = 47.066876606;
        double r18214642 = r18214640 + r18214641;
        double r18214643 = r18214631 / r18214642;
        return r18214643;
}
↓
double f(double x, double y, double z) {
        double r18214644 = x;
        double r18214645 = -5.291917632680427e+28;
        bool r18214646 = r18214644 <= r18214645;
        double r18214647 = 2.0;
        double r18214648 = r18214644 - r18214647;
        double r18214649 = 4.16438922228;
        double r18214650 = 101.7851458539211;
        double r18214651 = r18214650 / r18214644;
        double r18214652 = r18214649 - r18214651;
        double r18214653 = y;
        double r18214654 = r18214644 * r18214644;
        double r18214655 = r18214644 * r18214654;
        double r18214656 = r18214653 / r18214655;
        double r18214657 = r18214652 + r18214656;
        double r18214658 = r18214648 * r18214657;
        double r18214659 = 1498395052788.919;
        bool r18214660 = r18214644 <= r18214659;
        double r18214661 = 137.519416416;
        double r18214662 = 78.6994924154;
        double r18214663 = r18214649 * r18214644;
        double r18214664 = r18214662 + r18214663;
        double r18214665 = r18214664 * r18214644;
        double r18214666 = r18214661 + r18214665;
        double r18214667 = r18214644 * r18214666;
        double r18214668 = r18214667 + r18214653;
        double r18214669 = r18214644 * r18214668;
        double r18214670 = z;
        double r18214671 = r18214669 + r18214670;
        double r18214672 = r18214647 * r18214647;
        double r18214673 = r18214654 - r18214672;
        double r18214674 = r18214671 * r18214673;
        double r18214675 = r18214647 + r18214644;
        double r18214676 = 43.3400022514;
        double r18214677 = r18214676 + r18214644;
        double r18214678 = r18214677 * r18214644;
        double r18214679 = 263.505074721;
        double r18214680 = r18214678 + r18214679;
        double r18214681 = r18214680 * r18214644;
        double r18214682 = 313.399215894;
        double r18214683 = r18214681 + r18214682;
        double r18214684 = r18214683 * r18214644;
        double r18214685 = 47.066876606;
        double r18214686 = r18214684 + r18214685;
        double r18214687 = r18214675 * r18214686;
        double r18214688 = r18214674 / r18214687;
        double r18214689 = 110.1139242984811;
        double r18214690 = r18214663 - r18214689;
        double r18214691 = r18214653 / r18214654;
        double r18214692 = r18214690 + r18214691;
        double r18214693 = r18214660 ? r18214688 : r18214692;
        double r18214694 = r18214646 ? r18214658 : r18214693;
        return r18214694;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.6
Target0.5
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.291917632680427e+28

    1. Initial program 58.4

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity58.4

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right)}}\]

    4. Applied times-frac54.1

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

    5. Simplified54.1

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
    6. Using strategy rm

    7. Applied clear-num54.1

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}}\]
    8. Using strategy rm

    9. Applied div-inv54.1

      \[\leadsto \left(x - 2\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right) \cdot \frac{1}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}}\]

    10. Taylor expanded around inf 1.5

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - 101.785145853921093817007204052060842514 \cdot \frac{1}{x}\right)}\]

    11. Simplified1.5

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\frac{y}{x \cdot \left(x \cdot x\right)} + \left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right)\right)}\]

    if -5.291917632680427e+28 < x < 1498395052788.919

    1. Initial program 0.3

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
    2. Using strategy rm

    3. Applied flip--0.4

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x - 2 \cdot 2}{x + 2}} \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    4. Applied associate-*l/0.5

      \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x - 2 \cdot 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{x + 2}}}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    5. Applied associate-/l/0.5

      \[\leadsto \color{blue}{\frac{\left(x \cdot x - 2 \cdot 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right) \cdot \left(x + 2\right)}}\]

    if 1498395052788.919 < x

    1. Initial program 54.5

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Taylor expanded around inf 2.3

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

    3. Simplified2.3

      \[\leadsto \color{blue}{\frac{y}{x \cdot x} + \left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -52919176326804272705496416256:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right) + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 1498395052788.9189453125:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot \left(137.5194164160000127594685181975364685059 + \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right) \cdot x\right) + y\right) + z\right) \cdot \left(x \cdot x - 2 \cdot 2\right)}{\left(2 + x\right) \cdot \left(\left(\left(\left(43.3400022514000013984514225739985704422 + x\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))