Average Error: 26.1 → 0.6
Time: 28.7s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.486655794493063927988268380417398027792 \cdot 10^{52}:\\ \;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \le 1.849715130765958258047258630293245781501 \cdot 10^{53}:\\ \;\;\;\;\frac{x - 2}{\frac{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}{z + x \cdot \left(y + \left(x \cdot \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right) + 137.5194164160000127594685181975364685059\right) \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\ \end{array}\]
\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 -1.486655794493063927988268380417398027792 \cdot 10^{52}:\\
\;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\

\mathbf{elif}\;x \le 1.849715130765958258047258630293245781501 \cdot 10^{53}:\\
\;\;\;\;\frac{x - 2}{\frac{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}{z + x \cdot \left(y + \left(x \cdot \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right) + 137.5194164160000127594685181975364685059\right) \cdot x\right)}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r22822510 = x;
        double r22822511 = 2.0;
        double r22822512 = r22822510 - r22822511;
        double r22822513 = 4.16438922228;
        double r22822514 = r22822510 * r22822513;
        double r22822515 = 78.6994924154;
        double r22822516 = r22822514 + r22822515;
        double r22822517 = r22822516 * r22822510;
        double r22822518 = 137.519416416;
        double r22822519 = r22822517 + r22822518;
        double r22822520 = r22822519 * r22822510;
        double r22822521 = y;
        double r22822522 = r22822520 + r22822521;
        double r22822523 = r22822522 * r22822510;
        double r22822524 = z;
        double r22822525 = r22822523 + r22822524;
        double r22822526 = r22822512 * r22822525;
        double r22822527 = 43.3400022514;
        double r22822528 = r22822510 + r22822527;
        double r22822529 = r22822528 * r22822510;
        double r22822530 = 263.505074721;
        double r22822531 = r22822529 + r22822530;
        double r22822532 = r22822531 * r22822510;
        double r22822533 = 313.399215894;
        double r22822534 = r22822532 + r22822533;
        double r22822535 = r22822534 * r22822510;
        double r22822536 = 47.066876606;
        double r22822537 = r22822535 + r22822536;
        double r22822538 = r22822526 / r22822537;
        return r22822538;
}
↓
double f(double x, double y, double z) {
        double r22822539 = x;
        double r22822540 = -1.486655794493064e+52;
        bool r22822541 = r22822539 <= r22822540;
        double r22822542 = 4.16438922228;
        double r22822543 = r22822542 * r22822539;
        double r22822544 = 110.1139242984811;
        double r22822545 = r22822543 - r22822544;
        double r22822546 = y;
        double r22822547 = r22822539 * r22822539;
        double r22822548 = r22822546 / r22822547;
        double r22822549 = r22822545 + r22822548;
        double r22822550 = 1.8497151307659583e+53;
        bool r22822551 = r22822539 <= r22822550;
        double r22822552 = 2.0;
        double r22822553 = r22822539 - r22822552;
        double r22822554 = 47.066876606;
        double r22822555 = 43.3400022514;
        double r22822556 = r22822555 + r22822539;
        double r22822557 = r22822539 * r22822556;
        double r22822558 = 263.505074721;
        double r22822559 = r22822557 + r22822558;
        double r22822560 = r22822539 * r22822559;
        double r22822561 = 313.399215894;
        double r22822562 = r22822560 + r22822561;
        double r22822563 = r22822539 * r22822562;
        double r22822564 = r22822554 + r22822563;
        double r22822565 = z;
        double r22822566 = 78.6994924154;
        double r22822567 = r22822566 + r22822543;
        double r22822568 = r22822539 * r22822567;
        double r22822569 = 137.519416416;
        double r22822570 = r22822568 + r22822569;
        double r22822571 = r22822570 * r22822539;
        double r22822572 = r22822546 + r22822571;
        double r22822573 = r22822539 * r22822572;
        double r22822574 = r22822565 + r22822573;
        double r22822575 = r22822564 / r22822574;
        double r22822576 = r22822553 / r22822575;
        double r22822577 = r22822551 ? r22822576 : r22822549;
        double r22822578 = r22822541 ? r22822549 : r22822577;
        return r22822578;
}

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

Original26.1
Target0.5
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 < -1.486655794493064e+52 or 1.8497151307659583e+53 < x

    1. Initial program 62.5

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

    if -1.486655794493064e+52 < x < 1.8497151307659583e+53

    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

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.486655794493063927988268380417398027792 \cdot 10^{52}:\\ \;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \le 1.849715130765958258047258630293245781501 \cdot 10^{53}:\\ \;\;\;\;\frac{x - 2}{\frac{47.06687660600000100430406746454536914825 + x \cdot \left(x \cdot \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) + 313.3992158940000081202015280723571777344\right)}{z + x \cdot \left(y + \left(x \cdot \left(78.69949241540000173245061887428164482117 + 4.16438922227999963610045597306452691555 \cdot x\right) + 137.5194164160000127594685181975364685059\right) \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810948655649553984403610229\right) + \frac{y}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))