Average Error: 25.9 → 0.9
Time: 33.8s
Precision: 64
\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.942009801436773 \cdot 10^{+25}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 3.788002828269881 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(z + \left(x \cdot \left(x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right) + 137.519416416\right) + y\right) \cdot x\right) \cdot \left(x - 2.0\right)}{47.066876606 + \left(\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514\right) \cdot x\right) - 263.505074721 \cdot 263.505074721\right) \cdot x}{\left(x + 43.3400022514\right) \cdot x - 263.505074721} + 313.399215894\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \end{array}\]
\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 -5.942009801436773 \cdot 10^{+25}:\\
\;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\

\mathbf{elif}\;x \le 3.788002828269881 \cdot 10^{+32}:\\
\;\;\;\;\frac{\left(z + \left(x \cdot \left(x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right) + 137.519416416\right) + y\right) \cdot x\right) \cdot \left(x - 2.0\right)}{47.066876606 + \left(\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514\right) \cdot x\right) - 263.505074721 \cdot 263.505074721\right) \cdot x}{\left(x + 43.3400022514\right) \cdot x - 263.505074721} + 313.399215894\right) \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r14297718 = x;
        double r14297719 = 2.0;
        double r14297720 = r14297718 - r14297719;
        double r14297721 = 4.16438922228;
        double r14297722 = r14297718 * r14297721;
        double r14297723 = 78.6994924154;
        double r14297724 = r14297722 + r14297723;
        double r14297725 = r14297724 * r14297718;
        double r14297726 = 137.519416416;
        double r14297727 = r14297725 + r14297726;
        double r14297728 = r14297727 * r14297718;
        double r14297729 = y;
        double r14297730 = r14297728 + r14297729;
        double r14297731 = r14297730 * r14297718;
        double r14297732 = z;
        double r14297733 = r14297731 + r14297732;
        double r14297734 = r14297720 * r14297733;
        double r14297735 = 43.3400022514;
        double r14297736 = r14297718 + r14297735;
        double r14297737 = r14297736 * r14297718;
        double r14297738 = 263.505074721;
        double r14297739 = r14297737 + r14297738;
        double r14297740 = r14297739 * r14297718;
        double r14297741 = 313.399215894;
        double r14297742 = r14297740 + r14297741;
        double r14297743 = r14297742 * r14297718;
        double r14297744 = 47.066876606;
        double r14297745 = r14297743 + r14297744;
        double r14297746 = r14297734 / r14297745;
        return r14297746;
}
↓
double f(double x, double y, double z) {
        double r14297747 = x;
        double r14297748 = -5.942009801436773e+25;
        bool r14297749 = r14297747 <= r14297748;
        double r14297750 = 4.16438922228;
        double r14297751 = r14297750 * r14297747;
        double r14297752 = y;
        double r14297753 = r14297747 * r14297747;
        double r14297754 = r14297752 / r14297753;
        double r14297755 = 110.1139242984811;
        double r14297756 = r14297754 - r14297755;
        double r14297757 = r14297751 + r14297756;
        double r14297758 = 3.788002828269881e+32;
        bool r14297759 = r14297747 <= r14297758;
        double r14297760 = z;
        double r14297761 = 78.6994924154;
        double r14297762 = r14297751 + r14297761;
        double r14297763 = r14297747 * r14297762;
        double r14297764 = 137.519416416;
        double r14297765 = r14297763 + r14297764;
        double r14297766 = r14297747 * r14297765;
        double r14297767 = r14297766 + r14297752;
        double r14297768 = r14297767 * r14297747;
        double r14297769 = r14297760 + r14297768;
        double r14297770 = 2.0;
        double r14297771 = r14297747 - r14297770;
        double r14297772 = r14297769 * r14297771;
        double r14297773 = 47.066876606;
        double r14297774 = 43.3400022514;
        double r14297775 = r14297747 + r14297774;
        double r14297776 = r14297775 * r14297747;
        double r14297777 = r14297776 * r14297776;
        double r14297778 = 263.505074721;
        double r14297779 = r14297778 * r14297778;
        double r14297780 = r14297777 - r14297779;
        double r14297781 = r14297780 * r14297747;
        double r14297782 = r14297776 - r14297778;
        double r14297783 = r14297781 / r14297782;
        double r14297784 = 313.399215894;
        double r14297785 = r14297783 + r14297784;
        double r14297786 = r14297785 * r14297747;
        double r14297787 = r14297773 + r14297786;
        double r14297788 = r14297772 / r14297787;
        double r14297789 = r14297759 ? r14297788 : r14297757;
        double r14297790 = r14297749 ? r14297757 : r14297789;
        return r14297790;
}

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

Original25.9
Target0.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.942009801436773e+25 or 3.788002828269881e+32 < x

    1. Initial program 56.3

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

    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

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

    if -5.942009801436773e+25 < x < 3.788002828269881e+32

    1. Initial program 0.5

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

    3. Applied flip-+0.5

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

    4. Applied associate-*l/0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.942009801436773 \cdot 10^{+25}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \mathbf{elif}\;x \le 3.788002828269881 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(z + \left(x \cdot \left(x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right) + 137.519416416\right) + y\right) \cdot x\right) \cdot \left(x - 2.0\right)}{47.066876606 + \left(\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514\right) \cdot x\right) - 263.505074721 \cdot 263.505074721\right) \cdot x}{\left(x + 43.3400022514\right) \cdot x - 263.505074721} + 313.399215894\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x + \left(\frac{y}{x \cdot x} - 110.1139242984811\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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)))