Average Error: 26.6 → 1.1
Time: 25.9s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -19101382361175614464448786333696:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \le 1498395052788.9189453125:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + \left(x \cdot \left(x \cdot \left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) + 137.5194164160000127594685181975364685059\right) + y\right) \cdot x\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{x \cdot x}\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]
\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 r15103205 = x;
        double r15103206 = 2.0;
        double r15103207 = r15103205 - r15103206;
        double r15103208 = 4.16438922228;
        double r15103209 = r15103205 * r15103208;
        double r15103210 = 78.6994924154;
        double r15103211 = r15103209 + r15103210;
        double r15103212 = r15103211 * r15103205;
        double r15103213 = 137.519416416;
        double r15103214 = r15103212 + r15103213;
        double r15103215 = r15103214 * r15103205;
        double r15103216 = y;
        double r15103217 = r15103215 + r15103216;
        double r15103218 = r15103217 * r15103205;
        double r15103219 = z;
        double r15103220 = r15103218 + r15103219;
        double r15103221 = r15103207 * r15103220;
        double r15103222 = 43.3400022514;
        double r15103223 = r15103205 + r15103222;
        double r15103224 = r15103223 * r15103205;
        double r15103225 = 263.505074721;
        double r15103226 = r15103224 + r15103225;
        double r15103227 = r15103226 * r15103205;
        double r15103228 = 313.399215894;
        double r15103229 = r15103227 + r15103228;
        double r15103230 = r15103229 * r15103205;
        double r15103231 = 47.066876606;
        double r15103232 = r15103230 + r15103231;
        double r15103233 = r15103221 / r15103232;
        return r15103233;
}
↓
double f(double x, double y, double z) {
        double r15103234 = x;
        double r15103235 = -1.9101382361175614e+31;
        bool r15103236 = r15103234 <= r15103235;
        double r15103237 = 4.16438922228;
        double r15103238 = y;
        double r15103239 = r15103234 * r15103234;
        double r15103240 = r15103238 / r15103239;
        double r15103241 = fma(r15103237, r15103234, r15103240);
        double r15103242 = 110.1139242984811;
        double r15103243 = r15103241 - r15103242;
        double r15103244 = 1498395052788.919;
        bool r15103245 = r15103234 <= r15103244;
        double r15103246 = 2.0;
        double r15103247 = r15103234 - r15103246;
        double r15103248 = z;
        double r15103249 = r15103237 * r15103234;
        double r15103250 = 78.6994924154;
        double r15103251 = r15103249 + r15103250;
        double r15103252 = r15103234 * r15103251;
        double r15103253 = 137.519416416;
        double r15103254 = r15103252 + r15103253;
        double r15103255 = r15103234 * r15103254;
        double r15103256 = r15103255 + r15103238;
        double r15103257 = r15103256 * r15103234;
        double r15103258 = r15103248 + r15103257;
        double r15103259 = r15103247 * r15103258;
        double r15103260 = 47.066876606;
        double r15103261 = 313.399215894;
        double r15103262 = 263.505074721;
        double r15103263 = 43.3400022514;
        double r15103264 = r15103234 + r15103263;
        double r15103265 = r15103264 * r15103234;
        double r15103266 = r15103262 + r15103265;
        double r15103267 = r15103234 * r15103266;
        double r15103268 = r15103261 + r15103267;
        double r15103269 = r15103234 * r15103268;
        double r15103270 = r15103260 + r15103269;
        double r15103271 = r15103259 / r15103270;
        double r15103272 = r15103245 ? r15103271 : r15103243;
        double r15103273 = r15103236 ? r15103243 : r15103272;
        return r15103273;
}

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)))