Average Error: 26.7 → 1.0
Time: 9.0s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -34042328811361574912 \lor \neg \left(x \le 1.079429159403752641178373627064126450738 \cdot 10^{44}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\frac{\left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) \cdot \left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) - 313.3992158940000081202015280723571777344 \cdot 313.3992158940000081202015280723571777344\right) \cdot x}{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x - 313.3992158940000081202015280723571777344} + 47.06687660600000100430406746454536914825}\\ \end{array}\]
\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 -34042328811361574912 \lor \neg \left(x \le 1.079429159403752641178373627064126450738 \cdot 10^{44}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\

\mathbf{else}:\\
\;\;\;\;\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)}{\frac{\left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) \cdot \left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) - 313.3992158940000081202015280723571777344 \cdot 313.3992158940000081202015280723571777344\right) \cdot x}{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x - 313.3992158940000081202015280723571777344} + 47.06687660600000100430406746454536914825}\\

\end{array}
double f(double x, double y, double z) {
        double r352492 = x;
        double r352493 = 2.0;
        double r352494 = r352492 - r352493;
        double r352495 = 4.16438922228;
        double r352496 = r352492 * r352495;
        double r352497 = 78.6994924154;
        double r352498 = r352496 + r352497;
        double r352499 = r352498 * r352492;
        double r352500 = 137.519416416;
        double r352501 = r352499 + r352500;
        double r352502 = r352501 * r352492;
        double r352503 = y;
        double r352504 = r352502 + r352503;
        double r352505 = r352504 * r352492;
        double r352506 = z;
        double r352507 = r352505 + r352506;
        double r352508 = r352494 * r352507;
        double r352509 = 43.3400022514;
        double r352510 = r352492 + r352509;
        double r352511 = r352510 * r352492;
        double r352512 = 263.505074721;
        double r352513 = r352511 + r352512;
        double r352514 = r352513 * r352492;
        double r352515 = 313.399215894;
        double r352516 = r352514 + r352515;
        double r352517 = r352516 * r352492;
        double r352518 = 47.066876606;
        double r352519 = r352517 + r352518;
        double r352520 = r352508 / r352519;
        return r352520;
}
↓
double f(double x, double y, double z) {
        double r352521 = x;
        double r352522 = -3.4042328811361575e+19;
        bool r352523 = r352521 <= r352522;
        double r352524 = 1.0794291594037526e+44;
        bool r352525 = r352521 <= r352524;
        double r352526 = !r352525;
        bool r352527 = r352523 || r352526;
        double r352528 = y;
        double r352529 = 2.0;
        double r352530 = pow(r352521, r352529);
        double r352531 = r352528 / r352530;
        double r352532 = 4.16438922228;
        double r352533 = r352532 * r352521;
        double r352534 = r352531 + r352533;
        double r352535 = 110.1139242984811;
        double r352536 = r352534 - r352535;
        double r352537 = 2.0;
        double r352538 = r352521 - r352537;
        double r352539 = r352521 * r352532;
        double r352540 = 78.6994924154;
        double r352541 = r352539 + r352540;
        double r352542 = r352541 * r352521;
        double r352543 = 137.519416416;
        double r352544 = r352542 + r352543;
        double r352545 = r352544 * r352521;
        double r352546 = r352545 + r352528;
        double r352547 = r352546 * r352521;
        double r352548 = z;
        double r352549 = r352547 + r352548;
        double r352550 = r352538 * r352549;
        double r352551 = 43.3400022514;
        double r352552 = r352521 + r352551;
        double r352553 = r352552 * r352521;
        double r352554 = 263.505074721;
        double r352555 = r352553 + r352554;
        double r352556 = r352555 * r352521;
        double r352557 = r352556 * r352556;
        double r352558 = 313.399215894;
        double r352559 = r352558 * r352558;
        double r352560 = r352557 - r352559;
        double r352561 = r352560 * r352521;
        double r352562 = r352556 - r352558;
        double r352563 = r352561 / r352562;
        double r352564 = 47.066876606;
        double r352565 = r352563 + r352564;
        double r352566 = r352550 / r352565;
        double r352567 = r352527 ? r352536 : r352566;
        return r352567;
}

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

Original26.7
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.4042328811361575e+19 or 1.0794291594037526e+44 < x

    1. Initial program 58.6

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

    2. Taylor expanded around inf 1.6

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

    if -3.4042328811361575e+19 < x < 1.0794291594037526e+44

    1. Initial program 0.6

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

    3. Applied flip-+0.6

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

    4. Applied associate-*l/0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -34042328811361574912 \lor \neg \left(x \le 1.079429159403752641178373627064126450738 \cdot 10^{44}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\frac{\left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) \cdot \left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x\right) - 313.3992158940000081202015280723571777344 \cdot 313.3992158940000081202015280723571777344\right) \cdot x}{\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x - 313.3992158940000081202015280723571777344} + 47.06687660600000100430406746454536914825}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.3261287258700048e62) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109) (if (< x 9.4299917145546727e55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z) (+ (* (+ (+ (* 263.50507472100003 x) (+ (* 43.3400022514000014 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606000001))) (- (+ (/ y (* x x)) (* 4.16438922227999964 x)) 110.11392429848109)))

  (/ (* (- x 2) (+ (* (+ (* (+ (* (+ (* x 4.16438922227999964) 78.6994924154000017) x) 137.51941641600001) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514000014) x) 263.50507472100003) x) 313.399215894) x) 47.066876606000001)))