Average Error: 26.8 → 1.0
Time: 24.8s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -557231718650900352:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 1746149424175314114869089655301603328:\\ \;\;\;\;\frac{z + x \cdot \left(y + x \cdot \left(137.5194164160000127594685181975364685059 + x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right)\right)\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(\frac{x \cdot \left(\left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot 263.5050747210000281484099105000495910645 + \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(\left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right)\right)}{\left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645 - \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot 263.5050747210000281484099105000495910645\right) + \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)} + 313.3992158940000081202015280723571777344\right)} \cdot \left(x - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right) + \frac{y}{x \cdot \left(x \cdot x\right)}\right) \cdot \left(x - 2\right)\\ \end{array}\]
\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 -557231718650900352:\\
\;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)\\

\mathbf{elif}\;x \le 1746149424175314114869089655301603328:\\
\;\;\;\;\frac{z + x \cdot \left(y + x \cdot \left(137.5194164160000127594685181975364685059 + x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right)\right)\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(\frac{x \cdot \left(\left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot 263.5050747210000281484099105000495910645 + \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(\left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right)\right)}{\left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645 - \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot 263.5050747210000281484099105000495910645\right) + \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)} + 313.3992158940000081202015280723571777344\right)} \cdot \left(x - 2\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right) + \frac{y}{x \cdot \left(x \cdot x\right)}\right) \cdot \left(x - 2\right)\\

\end{array}
double f(double x, double y, double z) {
        double r19279681 = x;
        double r19279682 = 2.0;
        double r19279683 = r19279681 - r19279682;
        double r19279684 = 4.16438922228;
        double r19279685 = r19279681 * r19279684;
        double r19279686 = 78.6994924154;
        double r19279687 = r19279685 + r19279686;
        double r19279688 = r19279687 * r19279681;
        double r19279689 = 137.519416416;
        double r19279690 = r19279688 + r19279689;
        double r19279691 = r19279690 * r19279681;
        double r19279692 = y;
        double r19279693 = r19279691 + r19279692;
        double r19279694 = r19279693 * r19279681;
        double r19279695 = z;
        double r19279696 = r19279694 + r19279695;
        double r19279697 = r19279683 * r19279696;
        double r19279698 = 43.3400022514;
        double r19279699 = r19279681 + r19279698;
        double r19279700 = r19279699 * r19279681;
        double r19279701 = 263.505074721;
        double r19279702 = r19279700 + r19279701;
        double r19279703 = r19279702 * r19279681;
        double r19279704 = 313.399215894;
        double r19279705 = r19279703 + r19279704;
        double r19279706 = r19279705 * r19279681;
        double r19279707 = 47.066876606;
        double r19279708 = r19279706 + r19279707;
        double r19279709 = r19279697 / r19279708;
        return r19279709;
}
↓
double f(double x, double y, double z) {
        double r19279710 = x;
        double r19279711 = -5.5723171865090035e+17;
        bool r19279712 = r19279710 <= r19279711;
        double r19279713 = 4.16438922228;
        double r19279714 = r19279713 * r19279710;
        double r19279715 = y;
        double r19279716 = r19279715 / r19279710;
        double r19279717 = r19279716 / r19279710;
        double r19279718 = 110.1139242984811;
        double r19279719 = r19279717 - r19279718;
        double r19279720 = r19279714 + r19279719;
        double r19279721 = 1.746149424175314e+36;
        bool r19279722 = r19279710 <= r19279721;
        double r19279723 = z;
        double r19279724 = 137.519416416;
        double r19279725 = 78.6994924154;
        double r19279726 = r19279714 + r19279725;
        double r19279727 = r19279710 * r19279726;
        double r19279728 = r19279724 + r19279727;
        double r19279729 = r19279710 * r19279728;
        double r19279730 = r19279715 + r19279729;
        double r19279731 = r19279710 * r19279730;
        double r19279732 = r19279723 + r19279731;
        double r19279733 = 47.066876606;
        double r19279734 = 263.505074721;
        double r19279735 = r19279734 * r19279734;
        double r19279736 = r19279735 * r19279734;
        double r19279737 = 43.3400022514;
        double r19279738 = r19279737 + r19279710;
        double r19279739 = r19279710 * r19279738;
        double r19279740 = r19279739 * r19279739;
        double r19279741 = r19279739 * r19279740;
        double r19279742 = r19279736 + r19279741;
        double r19279743 = r19279710 * r19279742;
        double r19279744 = r19279739 * r19279734;
        double r19279745 = r19279735 - r19279744;
        double r19279746 = r19279745 + r19279740;
        double r19279747 = r19279743 / r19279746;
        double r19279748 = 313.399215894;
        double r19279749 = r19279747 + r19279748;
        double r19279750 = r19279710 * r19279749;
        double r19279751 = r19279733 + r19279750;
        double r19279752 = r19279732 / r19279751;
        double r19279753 = 2.0;
        double r19279754 = r19279710 - r19279753;
        double r19279755 = r19279752 * r19279754;
        double r19279756 = 101.7851458539211;
        double r19279757 = r19279756 / r19279710;
        double r19279758 = r19279713 - r19279757;
        double r19279759 = r19279710 * r19279710;
        double r19279760 = r19279710 * r19279759;
        double r19279761 = r19279715 / r19279760;
        double r19279762 = r19279758 + r19279761;
        double r19279763 = r19279762 * r19279754;
        double r19279764 = r19279722 ? r19279755 : r19279763;
        double r19279765 = r19279712 ? r19279720 : r19279764;
        return r19279765;
}

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

Original26.8
Target0.6
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.5723171865090035e+17

    1. Initial program 56.7

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

    2. Taylor expanded around inf 2.2

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

    3. Simplified2.2

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

    if -5.5723171865090035e+17 < x < 1.746149424175314e+36

    1. Initial program 0.6

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

    3. Applied *-un-lft-identity0.6

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

    7. Applied flip3-+0.3

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\frac{{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)}^{3} + {263.5050747210000281484099105000495910645}^{3}}{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) + \left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645 - \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot 263.5050747210000281484099105000495910645\right)}} \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    8. Applied associate-*l/0.3

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\frac{\left({\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)}^{3} + {263.5050747210000281484099105000495910645}^{3}\right) \cdot x}{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) + \left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645 - \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot 263.5050747210000281484099105000495910645\right)}} + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    9. Simplified0.3

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

    if 1.746149424175314e+36 < x

    1. Initial program 59.2

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

    3. Applied *-un-lft-identity59.2

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

    4. Applied times-frac55.4

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

    5. Simplified55.4

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

    7. Applied div-inv55.4

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

    8. Taylor expanded around inf 1.4

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

    9. Simplified1.4

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\frac{y}{\left(x \cdot x\right) \cdot x} + \left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -557231718650900352:\\ \;\;\;\;4.16438922227999963610045597306452691555 \cdot x + \left(\frac{\frac{y}{x}}{x} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 1746149424175314114869089655301603328:\\ \;\;\;\;\frac{z + x \cdot \left(y + x \cdot \left(137.5194164160000127594685181975364685059 + x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right)\right)\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(\frac{x \cdot \left(\left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot 263.5050747210000281484099105000495910645 + \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(\left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)\right)\right)}{\left(263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645 - \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot 263.5050747210000281484099105000495910645\right) + \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right) \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right)\right)} + 313.3992158940000081202015280723571777344\right)} \cdot \left(x - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(4.16438922227999963610045597306452691555 - \frac{101.785145853921093817007204052060842514}{x}\right) + \frac{y}{x \cdot \left(x \cdot x\right)}\right) \cdot \left(x - 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))