Average Error: 26.8 → 1.0
Time: 11.1s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5554983803297941420000 \lor \neg \left(x \le 1.39857869664759388 \cdot 10^{51}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5554983803297941420000 \lor \neg \left(x \le 1.39857869664759388 \cdot 10^{51}\right):\\
\;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\

\mathbf{else}:\\
\;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\\

\end{array}
double f(double x, double y, double z) {
        double r452351 = x;
        double r452352 = 2.0;
        double r452353 = r452351 - r452352;
        double r452354 = 4.16438922228;
        double r452355 = r452351 * r452354;
        double r452356 = 78.6994924154;
        double r452357 = r452355 + r452356;
        double r452358 = r452357 * r452351;
        double r452359 = 137.519416416;
        double r452360 = r452358 + r452359;
        double r452361 = r452360 * r452351;
        double r452362 = y;
        double r452363 = r452361 + r452362;
        double r452364 = r452363 * r452351;
        double r452365 = z;
        double r452366 = r452364 + r452365;
        double r452367 = r452353 * r452366;
        double r452368 = 43.3400022514;
        double r452369 = r452351 + r452368;
        double r452370 = r452369 * r452351;
        double r452371 = 263.505074721;
        double r452372 = r452370 + r452371;
        double r452373 = r452372 * r452351;
        double r452374 = 313.399215894;
        double r452375 = r452373 + r452374;
        double r452376 = r452375 * r452351;
        double r452377 = 47.066876606;
        double r452378 = r452376 + r452377;
        double r452379 = r452367 / r452378;
        return r452379;
}
↓
double f(double x, double y, double z) {
        double r452380 = x;
        double r452381 = -5.554983803297941e+21;
        bool r452382 = r452380 <= r452381;
        double r452383 = 1.3985786966475939e+51;
        bool r452384 = r452380 <= r452383;
        double r452385 = !r452384;
        bool r452386 = r452382 || r452385;
        double r452387 = y;
        double r452388 = 2.0;
        double r452389 = pow(r452380, r452388);
        double r452390 = r452387 / r452389;
        double r452391 = 4.16438922228;
        double r452392 = r452391 * r452380;
        double r452393 = r452390 + r452392;
        double r452394 = 110.1139242984811;
        double r452395 = r452393 - r452394;
        double r452396 = 2.0;
        double r452397 = r452380 - r452396;
        double r452398 = 43.3400022514;
        double r452399 = r452380 + r452398;
        double r452400 = r452399 * r452380;
        double r452401 = 263.505074721;
        double r452402 = r452400 + r452401;
        double r452403 = r452402 * r452380;
        double r452404 = 313.399215894;
        double r452405 = r452403 + r452404;
        double r452406 = r452405 * r452380;
        double r452407 = 47.066876606;
        double r452408 = r452406 + r452407;
        double r452409 = sqrt(r452408);
        double r452410 = r452397 / r452409;
        double r452411 = r452380 * r452391;
        double r452412 = 78.6994924154;
        double r452413 = r452411 + r452412;
        double r452414 = r452413 * r452380;
        double r452415 = 137.519416416;
        double r452416 = r452414 + r452415;
        double r452417 = r452416 * r452380;
        double r452418 = r452417 + r452387;
        double r452419 = r452418 * r452380;
        double r452420 = z;
        double r452421 = r452419 + r452420;
        double r452422 = r452421 / r452409;
        double r452423 = r452410 * r452422;
        double r452424 = r452386 ? r452395 : r452423;
        return r452424;
}

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

Original26.8
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.554983803297941e+21 or 1.3985786966475939e+51 < x

    1. Initial program 59.5

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    2. Taylor expanded around inf 1.3

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

    if -5.554983803297941e+21 < x < 1.3985786966475939e+51

    1. Initial program 0.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt1.0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001} \cdot \sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}}\]

    4. Applied times-frac0.8

      \[\leadsto \color{blue}{\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5554983803297941420000 \lor \neg \left(x \le 1.39857869664759388 \cdot 10^{51}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\sqrt{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* 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) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))