Average Error: 27.4 → 0.5
Time: 59.9s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.348224008252599505987937867651156039734 \cdot 10^{55}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \le 7.788614813457574873741915444049682186916 \cdot 10^{59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]
\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 -2.348224008252599505987937867651156039734 \cdot 10^{55}:\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\

\mathbf{elif}\;x \le 7.788614813457574873741915444049682186916 \cdot 10^{59}:\\
\;\;\;\;\frac{\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)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r17102409 = x;
        double r17102410 = 2.0;
        double r17102411 = r17102409 - r17102410;
        double r17102412 = 4.16438922228;
        double r17102413 = r17102409 * r17102412;
        double r17102414 = 78.6994924154;
        double r17102415 = r17102413 + r17102414;
        double r17102416 = r17102415 * r17102409;
        double r17102417 = 137.519416416;
        double r17102418 = r17102416 + r17102417;
        double r17102419 = r17102418 * r17102409;
        double r17102420 = y;
        double r17102421 = r17102419 + r17102420;
        double r17102422 = r17102421 * r17102409;
        double r17102423 = z;
        double r17102424 = r17102422 + r17102423;
        double r17102425 = r17102411 * r17102424;
        double r17102426 = 43.3400022514;
        double r17102427 = r17102409 + r17102426;
        double r17102428 = r17102427 * r17102409;
        double r17102429 = 263.505074721;
        double r17102430 = r17102428 + r17102429;
        double r17102431 = r17102430 * r17102409;
        double r17102432 = 313.399215894;
        double r17102433 = r17102431 + r17102432;
        double r17102434 = r17102433 * r17102409;
        double r17102435 = 47.066876606;
        double r17102436 = r17102434 + r17102435;
        double r17102437 = r17102425 / r17102436;
        return r17102437;
}
↓
double f(double x, double y, double z) {
        double r17102438 = x;
        double r17102439 = -2.3482240082525995e+55;
        bool r17102440 = r17102438 <= r17102439;
        double r17102441 = 4.16438922228;
        double r17102442 = y;
        double r17102443 = r17102438 * r17102438;
        double r17102444 = r17102442 / r17102443;
        double r17102445 = fma(r17102438, r17102441, r17102444);
        double r17102446 = 110.1139242984811;
        double r17102447 = r17102445 - r17102446;
        double r17102448 = 7.788614813457575e+59;
        bool r17102449 = r17102438 <= r17102448;
        double r17102450 = 78.6994924154;
        double r17102451 = fma(r17102438, r17102441, r17102450);
        double r17102452 = 137.519416416;
        double r17102453 = fma(r17102438, r17102451, r17102452);
        double r17102454 = fma(r17102438, r17102453, r17102442);
        double r17102455 = z;
        double r17102456 = fma(r17102438, r17102454, r17102455);
        double r17102457 = 43.3400022514;
        double r17102458 = r17102438 + r17102457;
        double r17102459 = 263.505074721;
        double r17102460 = fma(r17102458, r17102438, r17102459);
        double r17102461 = 313.399215894;
        double r17102462 = fma(r17102460, r17102438, r17102461);
        double r17102463 = 47.066876606;
        double r17102464 = fma(r17102462, r17102438, r17102463);
        double r17102465 = 2.0;
        double r17102466 = r17102438 - r17102465;
        double r17102467 = r17102464 / r17102466;
        double r17102468 = r17102456 / r17102467;
        double r17102469 = r17102449 ? r17102468 : r17102447;
        double r17102470 = r17102440 ? r17102447 : r17102469;
        return r17102470;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -2.3482240082525995e+55 or 7.788614813457575e+59 < x

    1. Initial program 63.4

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

    2. Simplified59.6

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

    3. Taylor expanded around inf 0.4

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

    4. Simplified0.4

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

    if -2.3482240082525995e+55 < x < 7.788614813457575e+59

    1. Initial program 1.5

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

    2. Simplified0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.348224008252599505987937867651156039734 \cdot 10^{55}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \le 7.788614813457574873741915444049682186916 \cdot 10^{59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000013984514225739985704422, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))