Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 27.1 → 0.9

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.752871306409339616131638108606681891161 \cdot 10^{47} \lor \neg \left(x \le 1.737590974376384239355797117310707632004 \cdot 10^{47}\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - 101.785145853921093817007204052060842514 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x} + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r415793 = x;
        double r415794 = 2.0;
        double r415795 = r415793 - r415794;
        double r415796 = 4.16438922228;
        double r415797 = r415793 * r415796;
        double r415798 = 78.6994924154;
        double r415799 = r415797 + r415798;
        double r415800 = r415799 * r415793;
        double r415801 = 137.519416416;
        double r415802 = r415800 + r415801;
        double r415803 = r415802 * r415793;
        double r415804 = y;
        double r415805 = r415803 + r415804;
        double r415806 = r415805 * r415793;
        double r415807 = z;
        double r415808 = r415806 + r415807;
        double r415809 = r415795 * r415808;
        double r415810 = 43.3400022514;
        double r415811 = r415793 + r415810;
        double r415812 = r415811 * r415793;
        double r415813 = 263.505074721;
        double r415814 = r415812 + r415813;
        double r415815 = r415814 * r415793;
        double r415816 = 313.399215894;
        double r415817 = r415815 + r415816;
        double r415818 = r415817 * r415793;
        double r415819 = 47.066876606;
        double r415820 = r415818 + r415819;
        double r415821 = r415809 / r415820;
        return r415821;
}

↓

double f(double x, double y, double z) {
        double r415822 = x;
        double r415823 = -8.75287130640934e+47;
        bool r415824 = r415822 <= r415823;
        double r415825 = 1.7375909743763842e+47;
        bool r415826 = r415822 <= r415825;
        double r415827 = !r415826;
        bool r415828 = r415824 || r415827;
        double r415829 = 2.0;
        double r415830 = r415822 - r415829;
        double r415831 = y;
        double r415832 = 3.0;
        double r415833 = pow(r415822, r415832);
        double r415834 = r415831 / r415833;
        double r415835 = 4.16438922228;
        double r415836 = r415834 + r415835;
        double r415837 = 101.7851458539211;
        double r415838 = 1.0;
        double r415839 = r415838 / r415822;
        double r415840 = r415837 * r415839;
        double r415841 = r415836 - r415840;
        double r415842 = r415830 * r415841;
        double r415843 = r415822 * r415835;
        double r415844 = 78.6994924154;
        double r415845 = r415843 + r415844;
        double r415846 = r415845 * r415822;
        double r415847 = 137.519416416;
        double r415848 = r415846 + r415847;
        double r415849 = r415848 * r415822;
        double r415850 = r415849 + r415831;
        double r415851 = cbrt(r415822);
        double r415852 = r415851 * r415851;
        double r415853 = r415850 * r415852;
        double r415854 = r415853 * r415851;
        double r415855 = z;
        double r415856 = r415854 + r415855;
        double r415857 = 43.3400022514;
        double r415858 = r415822 + r415857;
        double r415859 = r415858 * r415822;
        double r415860 = 263.505074721;
        double r415861 = r415859 + r415860;
        double r415862 = r415861 * r415822;
        double r415863 = 313.399215894;
        double r415864 = r415862 + r415863;
        double r415865 = r415864 * r415822;
        double r415866 = 47.066876606;
        double r415867 = r415865 + r415866;
        double r415868 = r415856 / r415867;
        double r415869 = r415830 * r415868;
        double r415870 = r415828 ? r415842 : r415869;
        return r415870;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.1
Target	0.6
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if x < -8.75287130640934e+47 or 1.7375909743763842e+47 < x

   1. Initial program 61.7

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

   3. Applied *-un-lft-identity 61.7

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

   4. Applied times-frac 57.5

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

   5. Simplified 57.5

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

   6. Taylor expanded around inf 0.9

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

   if -8.75287130640934e+47 < x < 1.7375909743763842e+47

   1. Initial program 1.0

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

   3. Applied *-un-lft-identity 1.0

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

   4. Applied times-frac 0.5

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

   5. Simplified 0.5

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

   7. Applied add-cube-cbrt 0.9

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

   8. Applied associate-*r* 0.9

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.752871306409339616131638108606681891161 \cdot 10^{47} \lor \neg \left(x \le 1.737590974376384239355797117310707632004 \cdot 10^{47}\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - 101.785145853921093817007204052060842514 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x} + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))