Average Error: 27.4 → 0.8
Time: 1.1m
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2968363930612989388200195091243076681728:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 2.02868606960421768723738828952845647677 \cdot 10^{61}:\\ \;\;\;\;\frac{x - 2}{\frac{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}{\frac{x \cdot \left(y + \left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x\right) + z}{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]
\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 -2968363930612989388200195091243076681728:\\
\;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\

\mathbf{elif}\;x \le 2.02868606960421768723738828952845647677 \cdot 10^{61}:\\
\;\;\;\;\frac{x - 2}{\frac{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}{\frac{x \cdot \left(y + \left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x\right) + z}{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\

\end{array}
double f(double x, double y, double z) {
        double r24540179 = x;
        double r24540180 = 2.0;
        double r24540181 = r24540179 - r24540180;
        double r24540182 = 4.16438922228;
        double r24540183 = r24540179 * r24540182;
        double r24540184 = 78.6994924154;
        double r24540185 = r24540183 + r24540184;
        double r24540186 = r24540185 * r24540179;
        double r24540187 = 137.519416416;
        double r24540188 = r24540186 + r24540187;
        double r24540189 = r24540188 * r24540179;
        double r24540190 = y;
        double r24540191 = r24540189 + r24540190;
        double r24540192 = r24540191 * r24540179;
        double r24540193 = z;
        double r24540194 = r24540192 + r24540193;
        double r24540195 = r24540181 * r24540194;
        double r24540196 = 43.3400022514;
        double r24540197 = r24540179 + r24540196;
        double r24540198 = r24540197 * r24540179;
        double r24540199 = 263.505074721;
        double r24540200 = r24540198 + r24540199;
        double r24540201 = r24540200 * r24540179;
        double r24540202 = 313.399215894;
        double r24540203 = r24540201 + r24540202;
        double r24540204 = r24540203 * r24540179;
        double r24540205 = 47.066876606;
        double r24540206 = r24540204 + r24540205;
        double r24540207 = r24540195 / r24540206;
        return r24540207;
}
↓
double f(double x, double y, double z) {
        double r24540208 = x;
        double r24540209 = -2.9683639306129894e+39;
        bool r24540210 = r24540208 <= r24540209;
        double r24540211 = y;
        double r24540212 = r24540208 * r24540208;
        double r24540213 = r24540211 / r24540212;
        double r24540214 = 4.16438922228;
        double r24540215 = r24540208 * r24540214;
        double r24540216 = 110.1139242984811;
        double r24540217 = r24540215 - r24540216;
        double r24540218 = r24540213 + r24540217;
        double r24540219 = 2.0286860696042177e+61;
        bool r24540220 = r24540208 <= r24540219;
        double r24540221 = 2.0;
        double r24540222 = r24540208 - r24540221;
        double r24540223 = 47.066876606;
        double r24540224 = 43.3400022514;
        double r24540225 = r24540224 + r24540208;
        double r24540226 = r24540208 * r24540225;
        double r24540227 = 263.505074721;
        double r24540228 = r24540226 + r24540227;
        double r24540229 = r24540208 * r24540228;
        double r24540230 = 313.399215894;
        double r24540231 = r24540229 + r24540230;
        double r24540232 = r24540208 * r24540231;
        double r24540233 = r24540223 + r24540232;
        double r24540234 = sqrt(r24540233);
        double r24540235 = 78.6994924154;
        double r24540236 = r24540215 + r24540235;
        double r24540237 = r24540236 * r24540208;
        double r24540238 = 137.519416416;
        double r24540239 = r24540237 + r24540238;
        double r24540240 = r24540239 * r24540208;
        double r24540241 = r24540211 + r24540240;
        double r24540242 = r24540208 * r24540241;
        double r24540243 = z;
        double r24540244 = r24540242 + r24540243;
        double r24540245 = r24540244 / r24540234;
        double r24540246 = r24540234 / r24540245;
        double r24540247 = r24540222 / r24540246;
        double r24540248 = r24540220 ? r24540247 : r24540218;
        double r24540249 = r24540210 ? r24540218 : r24540248;
        return r24540249;
}

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

Original27.4
Target0.5
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.9683639306129894e+39 or 2.0286860696042177e+61 < x

    1. Initial program 62.1

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

    3. Applied associate-/l*58.1

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

    5. Applied *-un-lft-identity58.1

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

    6. Applied add-sqr-sqrt58.1

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

    7. Applied times-frac58.1

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

    8. Applied associate-/r*58.1

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

    9. Taylor expanded around inf 0.7

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

    10. Simplified0.7

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

    if -2.9683639306129894e+39 < x < 2.0286860696042177e+61

    1. Initial program 1.3

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

    3. Applied associate-/l*0.7

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

    5. Applied add-sqr-sqrt0.8

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

    6. Applied associate-/l*0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2968363930612989388200195091243076681728:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 2.02868606960421768723738828952845647677 \cdot 10^{61}:\\ \;\;\;\;\frac{x - 2}{\frac{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}{\frac{x \cdot \left(y + \left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x\right) + z}{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))