Average Error: 26.6 → 1.1
Time: 25.5s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -19101382361175614464448786333696:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \le 1498395052788.9189453125:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + \left(x \cdot \left(x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) + 137.5194164160000127594685181975364685059\right) + y\right) \cdot x\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]
\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 -19101382361175614464448786333696:\\
\;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\

\mathbf{elif}\;x \le 1498395052788.9189453125:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + \left(x \cdot \left(x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) + 137.5194164160000127594685181975364685059\right) + y\right) \cdot x\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r15250601 = x;
        double r15250602 = 2.0;
        double r15250603 = r15250601 - r15250602;
        double r15250604 = 4.16438922228;
        double r15250605 = r15250601 * r15250604;
        double r15250606 = 78.6994924154;
        double r15250607 = r15250605 + r15250606;
        double r15250608 = r15250607 * r15250601;
        double r15250609 = 137.519416416;
        double r15250610 = r15250608 + r15250609;
        double r15250611 = r15250610 * r15250601;
        double r15250612 = y;
        double r15250613 = r15250611 + r15250612;
        double r15250614 = r15250613 * r15250601;
        double r15250615 = z;
        double r15250616 = r15250614 + r15250615;
        double r15250617 = r15250603 * r15250616;
        double r15250618 = 43.3400022514;
        double r15250619 = r15250601 + r15250618;
        double r15250620 = r15250619 * r15250601;
        double r15250621 = 263.505074721;
        double r15250622 = r15250620 + r15250621;
        double r15250623 = r15250622 * r15250601;
        double r15250624 = 313.399215894;
        double r15250625 = r15250623 + r15250624;
        double r15250626 = r15250625 * r15250601;
        double r15250627 = 47.066876606;
        double r15250628 = r15250626 + r15250627;
        double r15250629 = r15250617 / r15250628;
        return r15250629;
}
↓
double f(double x, double y, double z) {
        double r15250630 = x;
        double r15250631 = -1.9101382361175614e+31;
        bool r15250632 = r15250630 <= r15250631;
        double r15250633 = 4.16438922228;
        double r15250634 = y;
        double r15250635 = r15250630 * r15250630;
        double r15250636 = r15250634 / r15250635;
        double r15250637 = fma(r15250633, r15250630, r15250636);
        double r15250638 = 110.1139242984811;
        double r15250639 = r15250637 - r15250638;
        double r15250640 = 1498395052788.919;
        bool r15250641 = r15250630 <= r15250640;
        double r15250642 = 2.0;
        double r15250643 = r15250630 - r15250642;
        double r15250644 = z;
        double r15250645 = r15250633 * r15250630;
        double r15250646 = 78.6994924154;
        double r15250647 = r15250645 + r15250646;
        double r15250648 = r15250630 * r15250647;
        double r15250649 = 137.519416416;
        double r15250650 = r15250648 + r15250649;
        double r15250651 = r15250630 * r15250650;
        double r15250652 = r15250651 + r15250634;
        double r15250653 = r15250652 * r15250630;
        double r15250654 = r15250644 + r15250653;
        double r15250655 = r15250643 * r15250654;
        double r15250656 = 47.066876606;
        double r15250657 = 313.399215894;
        double r15250658 = 263.505074721;
        double r15250659 = 43.3400022514;
        double r15250660 = r15250630 + r15250659;
        double r15250661 = r15250660 * r15250630;
        double r15250662 = r15250658 + r15250661;
        double r15250663 = r15250630 * r15250662;
        double r15250664 = r15250657 + r15250663;
        double r15250665 = r15250630 * r15250664;
        double r15250666 = r15250656 + r15250665;
        double r15250667 = r15250655 / r15250666;
        double r15250668 = r15250641 ? r15250667 : r15250639;
        double r15250669 = r15250632 ? r15250639 : r15250668;
        return r15250669;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.6
Target0.5
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.9101382361175614e+31 or 1498395052788.919 < x

    1. Initial program 56.5

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 1.9

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

    4. Simplified1.9

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

    if -1.9101382361175614e+31 < x < 1498395052788.919

    1. Initial program 0.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -19101382361175614464448786333696:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \le 1498395052788.9189453125:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + \left(x \cdot \left(x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) + 137.5194164160000127594685181975364685059\right) + y\right) \cdot x\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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)))