Average Error: 26.0 → 0.8
Time: 25.1s
Precision: 64
\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.140336158145698 \cdot 10^{+17}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \le 1.692613215272597 \cdot 10^{+44}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \end{array}\]
\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.140336158145698 \cdot 10^{+17}:\\
\;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\

\mathbf{elif}\;x \le 1.692613215272597 \cdot 10^{+44}:\\
\;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r21774703 = x;
        double r21774704 = 2.0;
        double r21774705 = r21774703 - r21774704;
        double r21774706 = 4.16438922228;
        double r21774707 = r21774703 * r21774706;
        double r21774708 = 78.6994924154;
        double r21774709 = r21774707 + r21774708;
        double r21774710 = r21774709 * r21774703;
        double r21774711 = 137.519416416;
        double r21774712 = r21774710 + r21774711;
        double r21774713 = r21774712 * r21774703;
        double r21774714 = y;
        double r21774715 = r21774713 + r21774714;
        double r21774716 = r21774715 * r21774703;
        double r21774717 = z;
        double r21774718 = r21774716 + r21774717;
        double r21774719 = r21774705 * r21774718;
        double r21774720 = 43.3400022514;
        double r21774721 = r21774703 + r21774720;
        double r21774722 = r21774721 * r21774703;
        double r21774723 = 263.505074721;
        double r21774724 = r21774722 + r21774723;
        double r21774725 = r21774724 * r21774703;
        double r21774726 = 313.399215894;
        double r21774727 = r21774725 + r21774726;
        double r21774728 = r21774727 * r21774703;
        double r21774729 = 47.066876606;
        double r21774730 = r21774728 + r21774729;
        double r21774731 = r21774719 / r21774730;
        return r21774731;
}
↓
double f(double x, double y, double z) {
        double r21774732 = x;
        double r21774733 = -5.140336158145698e+17;
        bool r21774734 = r21774732 <= r21774733;
        double r21774735 = 4.16438922228;
        double r21774736 = r21774735 * r21774732;
        double r21774737 = 110.1139242984811;
        double r21774738 = r21774736 - r21774737;
        double r21774739 = y;
        double r21774740 = r21774739 / r21774732;
        double r21774741 = r21774740 / r21774732;
        double r21774742 = r21774738 + r21774741;
        double r21774743 = 1.692613215272597e+44;
        bool r21774744 = r21774732 <= r21774743;
        double r21774745 = 2.0;
        double r21774746 = r21774732 - r21774745;
        double r21774747 = z;
        double r21774748 = 137.519416416;
        double r21774749 = 78.6994924154;
        double r21774750 = r21774736 + r21774749;
        double r21774751 = r21774732 * r21774750;
        double r21774752 = r21774748 + r21774751;
        double r21774753 = r21774732 * r21774752;
        double r21774754 = r21774739 + r21774753;
        double r21774755 = r21774732 * r21774754;
        double r21774756 = r21774747 + r21774755;
        double r21774757 = 47.066876606;
        double r21774758 = 313.399215894;
        double r21774759 = 263.505074721;
        double r21774760 = 43.3400022514;
        double r21774761 = r21774732 + r21774760;
        double r21774762 = r21774761 * r21774732;
        double r21774763 = r21774759 + r21774762;
        double r21774764 = r21774732 * r21774763;
        double r21774765 = r21774758 + r21774764;
        double r21774766 = r21774732 * r21774765;
        double r21774767 = r21774757 + r21774766;
        double r21774768 = r21774756 / r21774767;
        double r21774769 = r21774746 * r21774768;
        double r21774770 = r21774744 ? r21774769 : r21774742;
        double r21774771 = r21774734 ? r21774742 : r21774770;
        return r21774771;
}

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

Original26.0
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.140336158145698e+17 or 1.692613215272597e+44 < x

    1. Initial program 56.2

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

    2. Taylor expanded around inf 1.3

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922228 \cdot x\right) - 110.1139242984811}\]

    3. Simplified1.3

      \[\leadsto \color{blue}{\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}}\]

    if -5.140336158145698e+17 < x < 1.692613215272597e+44

    1. Initial program 0.7

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.7

      \[\leadsto \frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606\right)}}\]

    4. Applied times-frac0.3

      \[\leadsto \color{blue}{\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}\]

    5. Simplified0.3

      \[\leadsto \color{blue}{\left(x - 2.0\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.140336158145698 \cdot 10^{+17}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \le 1.692613215272597 \cdot 10^{+44}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + \left(x + 43.3400022514\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 \cdot x - 110.1139242984811\right) + \frac{\frac{y}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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) (/ (+ (* (+ (* (+ (* (+ (* 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)))