Average Error: 26.8 → 0.6
Time: 28.1s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.215856075377309154727911747734784083911 \cdot 10^{69}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{\frac{y}{x \cdot x}}{x} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \mathbf{elif}\;x \le 3.893369127510250731208425645412434685496 \cdot 10^{68}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{1}{\frac{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right)\right)}{z + \left(x \cdot \left(137.5194164160000127594685181975364685059 + \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) \cdot x\right) + y\right) \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{\frac{y}{x \cdot x}}{x} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \end{array}\]
\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 -1.215856075377309154727911747734784083911 \cdot 10^{69}:\\
\;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{\frac{y}{x \cdot x}}{x} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\

\mathbf{elif}\;x \le 3.893369127510250731208425645412434685496 \cdot 10^{68}:\\
\;\;\;\;\left(x - 2\right) \cdot \frac{1}{\frac{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right)\right)}{z + \left(x \cdot \left(137.5194164160000127594685181975364685059 + \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) \cdot x\right) + y\right) \cdot x}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r17114153 = x;
        double r17114154 = 2.0;
        double r17114155 = r17114153 - r17114154;
        double r17114156 = 4.16438922228;
        double r17114157 = r17114153 * r17114156;
        double r17114158 = 78.6994924154;
        double r17114159 = r17114157 + r17114158;
        double r17114160 = r17114159 * r17114153;
        double r17114161 = 137.519416416;
        double r17114162 = r17114160 + r17114161;
        double r17114163 = r17114162 * r17114153;
        double r17114164 = y;
        double r17114165 = r17114163 + r17114164;
        double r17114166 = r17114165 * r17114153;
        double r17114167 = z;
        double r17114168 = r17114166 + r17114167;
        double r17114169 = r17114155 * r17114168;
        double r17114170 = 43.3400022514;
        double r17114171 = r17114153 + r17114170;
        double r17114172 = r17114171 * r17114153;
        double r17114173 = 263.505074721;
        double r17114174 = r17114172 + r17114173;
        double r17114175 = r17114174 * r17114153;
        double r17114176 = 313.399215894;
        double r17114177 = r17114175 + r17114176;
        double r17114178 = r17114177 * r17114153;
        double r17114179 = 47.066876606;
        double r17114180 = r17114178 + r17114179;
        double r17114181 = r17114169 / r17114180;
        return r17114181;
}
↓
double f(double x, double y, double z) {
        double r17114182 = x;
        double r17114183 = -1.2158560753773092e+69;
        bool r17114184 = r17114182 <= r17114183;
        double r17114185 = 2.0;
        double r17114186 = r17114182 - r17114185;
        double r17114187 = y;
        double r17114188 = r17114182 * r17114182;
        double r17114189 = r17114187 / r17114188;
        double r17114190 = r17114189 / r17114182;
        double r17114191 = 4.16438922228;
        double r17114192 = r17114190 + r17114191;
        double r17114193 = 101.7851458539211;
        double r17114194 = r17114193 / r17114182;
        double r17114195 = r17114192 - r17114194;
        double r17114196 = r17114186 * r17114195;
        double r17114197 = 3.8933691275102507e+68;
        bool r17114198 = r17114182 <= r17114197;
        double r17114199 = 1.0;
        double r17114200 = 47.066876606;
        double r17114201 = 313.399215894;
        double r17114202 = 43.3400022514;
        double r17114203 = r17114202 + r17114182;
        double r17114204 = r17114182 * r17114203;
        double r17114205 = 263.505074721;
        double r17114206 = r17114204 + r17114205;
        double r17114207 = r17114182 * r17114206;
        double r17114208 = r17114201 + r17114207;
        double r17114209 = r17114182 * r17114208;
        double r17114210 = r17114200 + r17114209;
        double r17114211 = z;
        double r17114212 = 137.519416416;
        double r17114213 = r17114191 * r17114182;
        double r17114214 = 78.6994924154;
        double r17114215 = r17114213 + r17114214;
        double r17114216 = r17114215 * r17114182;
        double r17114217 = r17114212 + r17114216;
        double r17114218 = r17114182 * r17114217;
        double r17114219 = r17114218 + r17114187;
        double r17114220 = r17114219 * r17114182;
        double r17114221 = r17114211 + r17114220;
        double r17114222 = r17114210 / r17114221;
        double r17114223 = r17114199 / r17114222;
        double r17114224 = r17114186 * r17114223;
        double r17114225 = r17114198 ? r17114224 : r17114196;
        double r17114226 = r17114184 ? r17114196 : r17114225;
        return r17114226;
}

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
Target0.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.2158560753773092e+69 or 3.8933691275102507e+68 < x

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity64.0

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

    4. Applied times-frac61.9

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

    5. Simplified61.9

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

    6. Taylor expanded around inf 0.1

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

    7. Simplified0.1

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

    if -1.2158560753773092e+69 < x < 3.8933691275102507e+68

    1. Initial program 2.9

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

    3. Applied *-un-lft-identity2.9

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

    4. Applied times-frac0.7

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

    5. Simplified0.7

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

    7. Applied clear-num0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.215856075377309154727911747734784083911 \cdot 10^{69}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{\frac{y}{x \cdot x}}{x} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \mathbf{elif}\;x \le 3.893369127510250731208425645412434685496 \cdot 10^{68}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{1}{\frac{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right)\right)}{z + \left(x \cdot \left(137.5194164160000127594685181975364685059 + \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) \cdot x\right) + y\right) \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{\frac{y}{x \cdot x}}{x} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))