Average Error: 27.4 → 0.8
Time: 1.0m
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2968363930612989388200195091243076681728:\\ \;\;\;\;\frac{\frac{y}{x}}{x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\ \mathbf{elif}\;x \le 2.02868606960421768723738828952845647677 \cdot 10^{61}:\\ \;\;\;\;\frac{x - 2}{\frac{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}{\frac{x \cdot \left(y + \left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x\right) + z}{\sqrt{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\ \end{array}\]
\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{\frac{y}{x}}{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{\frac{y}{x}}{x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810948655649553984403610229\right)\\

\end{array}
double f(double x, double y, double z) {
        double r27690293 = x;
        double r27690294 = 2.0;
        double r27690295 = r27690293 - r27690294;
        double r27690296 = 4.16438922228;
        double r27690297 = r27690293 * r27690296;
        double r27690298 = 78.6994924154;
        double r27690299 = r27690297 + r27690298;
        double r27690300 = r27690299 * r27690293;
        double r27690301 = 137.519416416;
        double r27690302 = r27690300 + r27690301;
        double r27690303 = r27690302 * r27690293;
        double r27690304 = y;
        double r27690305 = r27690303 + r27690304;
        double r27690306 = r27690305 * r27690293;
        double r27690307 = z;
        double r27690308 = r27690306 + r27690307;
        double r27690309 = r27690295 * r27690308;
        double r27690310 = 43.3400022514;
        double r27690311 = r27690293 + r27690310;
        double r27690312 = r27690311 * r27690293;
        double r27690313 = 263.505074721;
        double r27690314 = r27690312 + r27690313;
        double r27690315 = r27690314 * r27690293;
        double r27690316 = 313.399215894;
        double r27690317 = r27690315 + r27690316;
        double r27690318 = r27690317 * r27690293;
        double r27690319 = 47.066876606;
        double r27690320 = r27690318 + r27690319;
        double r27690321 = r27690309 / r27690320;
        return r27690321;
}
↓
double f(double x, double y, double z) {
        double r27690322 = x;
        double r27690323 = -2.9683639306129894e+39;
        bool r27690324 = r27690322 <= r27690323;
        double r27690325 = y;
        double r27690326 = r27690325 / r27690322;
        double r27690327 = r27690326 / r27690322;
        double r27690328 = 4.16438922228;
        double r27690329 = r27690322 * r27690328;
        double r27690330 = 110.1139242984811;
        double r27690331 = r27690329 - r27690330;
        double r27690332 = r27690327 + r27690331;
        double r27690333 = 2.0286860696042177e+61;
        bool r27690334 = r27690322 <= r27690333;
        double r27690335 = 2.0;
        double r27690336 = r27690322 - r27690335;
        double r27690337 = 47.066876606;
        double r27690338 = 43.3400022514;
        double r27690339 = r27690338 + r27690322;
        double r27690340 = r27690322 * r27690339;
        double r27690341 = 263.505074721;
        double r27690342 = r27690340 + r27690341;
        double r27690343 = r27690322 * r27690342;
        double r27690344 = 313.399215894;
        double r27690345 = r27690343 + r27690344;
        double r27690346 = r27690322 * r27690345;
        double r27690347 = r27690337 + r27690346;
        double r27690348 = sqrt(r27690347);
        double r27690349 = 78.6994924154;
        double r27690350 = r27690329 + r27690349;
        double r27690351 = r27690350 * r27690322;
        double r27690352 = 137.519416416;
        double r27690353 = r27690351 + r27690352;
        double r27690354 = r27690353 * r27690322;
        double r27690355 = r27690325 + r27690354;
        double r27690356 = r27690322 * r27690355;
        double r27690357 = z;
        double r27690358 = r27690356 + r27690357;
        double r27690359 = r27690358 / r27690348;
        double r27690360 = r27690348 / r27690359;
        double r27690361 = r27690336 / r27690360;
        double r27690362 = r27690334 ? r27690361 : r27690332;
        double r27690363 = r27690324 ? r27690332 : r27690362;
        return r27690363;
}

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. Taylor expanded around inf 0.7

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

    5. Simplified0.7

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

    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

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

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)))