Average Error: 26.0 → 1.0
Time: 10.2s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6161725570264887795601484283904 \lor \neg \left(x \le 2.405459444742494719487876149488306641587 \cdot 10^{53}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\left(\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right) - 78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117\right) \cdot x}{x \cdot 4.16438922227999963610045597306452691555 - 78.69949241540000173245061887428164482117} + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]
\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 -6161725570264887795601484283904 \lor \neg \left(x \le 2.405459444742494719487876149488306641587 \cdot 10^{53}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\left(\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right) - 78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117\right) \cdot x}{x \cdot 4.16438922227999963610045597306452691555 - 78.69949241540000173245061887428164482117} + 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}\\

\end{array}
double f(double x, double y, double z) {
        double r606742 = x;
        double r606743 = 2.0;
        double r606744 = r606742 - r606743;
        double r606745 = 4.16438922228;
        double r606746 = r606742 * r606745;
        double r606747 = 78.6994924154;
        double r606748 = r606746 + r606747;
        double r606749 = r606748 * r606742;
        double r606750 = 137.519416416;
        double r606751 = r606749 + r606750;
        double r606752 = r606751 * r606742;
        double r606753 = y;
        double r606754 = r606752 + r606753;
        double r606755 = r606754 * r606742;
        double r606756 = z;
        double r606757 = r606755 + r606756;
        double r606758 = r606744 * r606757;
        double r606759 = 43.3400022514;
        double r606760 = r606742 + r606759;
        double r606761 = r606760 * r606742;
        double r606762 = 263.505074721;
        double r606763 = r606761 + r606762;
        double r606764 = r606763 * r606742;
        double r606765 = 313.399215894;
        double r606766 = r606764 + r606765;
        double r606767 = r606766 * r606742;
        double r606768 = 47.066876606;
        double r606769 = r606767 + r606768;
        double r606770 = r606758 / r606769;
        return r606770;
}
↓
double f(double x, double y, double z) {
        double r606771 = x;
        double r606772 = -6.161725570264888e+30;
        bool r606773 = r606771 <= r606772;
        double r606774 = 2.4054594447424947e+53;
        bool r606775 = r606771 <= r606774;
        double r606776 = !r606775;
        bool r606777 = r606773 || r606776;
        double r606778 = y;
        double r606779 = 2.0;
        double r606780 = pow(r606771, r606779);
        double r606781 = r606778 / r606780;
        double r606782 = 4.16438922228;
        double r606783 = r606782 * r606771;
        double r606784 = r606781 + r606783;
        double r606785 = 110.1139242984811;
        double r606786 = r606784 - r606785;
        double r606787 = 2.0;
        double r606788 = r606771 - r606787;
        double r606789 = r606771 * r606782;
        double r606790 = r606789 * r606789;
        double r606791 = 78.6994924154;
        double r606792 = r606791 * r606791;
        double r606793 = r606790 - r606792;
        double r606794 = r606793 * r606771;
        double r606795 = r606789 - r606791;
        double r606796 = r606794 / r606795;
        double r606797 = 137.519416416;
        double r606798 = r606796 + r606797;
        double r606799 = r606798 * r606771;
        double r606800 = r606799 + r606778;
        double r606801 = r606800 * r606771;
        double r606802 = z;
        double r606803 = r606801 + r606802;
        double r606804 = r606788 * r606803;
        double r606805 = 43.3400022514;
        double r606806 = r606771 + r606805;
        double r606807 = r606806 * r606771;
        double r606808 = 263.505074721;
        double r606809 = r606807 + r606808;
        double r606810 = r606809 * r606771;
        double r606811 = 313.399215894;
        double r606812 = r606810 + r606811;
        double r606813 = r606812 * r606771;
        double r606814 = 47.066876606;
        double r606815 = r606813 + r606814;
        double r606816 = r606804 / r606815;
        double r606817 = r606777 ? r606786 : r606816;
        return r606817;
}

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
Target0.5
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 2 regimes

  2. if x < -6.161725570264888e+30 or 2.4054594447424947e+53 < x

    1. Initial program 60.0

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

    2. Taylor expanded around inf 1.0

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

    if -6.161725570264888e+30 < x < 2.4054594447424947e+53

    1. Initial program 1.1

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

    3. Applied flip-+1.1

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

    4. Applied associate-*l/1.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6161725570264887795601484283904 \lor \neg \left(x \le 2.405459444742494719487876149488306641587 \cdot 10^{53}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\left(\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right) - 78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117\right) \cdot x}{x \cdot 4.16438922227999963610045597306452691555 - 78.69949241540000173245061887428164482117} + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 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) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))