Average Error: 26.1 → 0.8
Time: 6.7s
Precision: binary64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -9.42722712682916 \cdot 10^{+65} \lor \neg \left(x \leq 3.110893828765909 \cdot 10^{+33}\right):\\ \;\;\;\;\left(x \cdot 4.16438922228 + \frac{y}{x \cdot x}\right) - 110.1139242984811\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 2}{\sqrt{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}} \cdot \left(x \cdot \left(y + x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right)\right) + z\right)}{\sqrt{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -9.42722712682916 \cdot 10^{+65} \lor \neg \left(x \leq 3.110893828765909 \cdot 10^{+33}\right):\\
\;\;\;\;\left(x \cdot 4.16438922228 + \frac{y}{x \cdot x}\right) - 110.1139242984811\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x - 2}{\sqrt{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}} \cdot \left(x \cdot \left(y + x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right)\right) + z\right)}{\sqrt{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}}\\

\end{array}
(FPCore (x y z)
 :precision binary64
 (/
  (*
   (- 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)))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.42722712682916e+65) (not (<= x 3.110893828765909e+33)))
   (- (+ (* x 4.16438922228) (/ y (* x x))) 110.1139242984811)
   (/
    (*
     (/
      (- x 2.0)
      (sqrt
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606)))
     (+
      (*
       x
       (+
        y
        (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))))
      z))
    (sqrt
     (+
      (* x (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
      47.066876606)))))
double code(double x, double y, double z) {
	return (((double) (((double) (x - 2.0)) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (x * 4.16438922228)) + 78.6994924154)) * x)) + 137.519416416)) * x)) + y)) * x)) + z)))) / ((double) (((double) (((double) (((double) (((double) (((double) (((double) (x + 43.3400022514)) * x)) + 263.505074721)) * x)) + 313.399215894)) * x)) + 47.066876606)));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (((x <= -9.42722712682916e+65) || !(x <= 3.110893828765909e+33))) {
		tmp = ((double) (((double) (((double) (x * 4.16438922228)) + (y / ((double) (x * x))))) - 110.1139242984811));
	} else {
		tmp = (((double) ((((double) (x - 2.0)) / ((double) sqrt(((double) (((double) (x * ((double) (((double) (x * ((double) (((double) (x * ((double) (x + 43.3400022514)))) + 263.505074721)))) + 313.399215894)))) + 47.066876606))))) * ((double) (((double) (x * ((double) (y + ((double) (x * ((double) (((double) (x * ((double) (((double) (x * 4.16438922228)) + 78.6994924154)))) + 137.519416416)))))))) + z)))) / ((double) sqrt(((double) (((double) (x * ((double) (((double) (x * ((double) (((double) (x * ((double) (x + 43.3400022514)))) + 263.505074721)))) + 313.399215894)))) + 47.066876606)))));
	}
	return tmp;
}

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

Original26.1
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -9.4272271268291591e65 or 3.1108938287659091e33 < x

    1. Initial program 61.3

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922228 \cdot x\right) - 110.1139242984811}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\left(x \cdot 4.16438922228 + \frac{y}{x \cdot x}\right) - 110.1139242984811}\]

    if -9.4272271268291591e65 < x < 3.1108938287659091e33

    1. Initial program 1.3

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary641.5

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \cdot \sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\]

    4. Applied associate-/r*_binary641.7

      \[\leadsto \color{blue}{\frac{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}}\]

    5. Simplified0.9

      \[\leadsto \frac{\color{blue}{\frac{x - 2}{\sqrt{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}}{\sqrt{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.42722712682916 \cdot 10^{+65} \lor \neg \left(x \leq 3.110893828765909 \cdot 10^{+33}\right):\\ \;\;\;\;\left(x \cdot 4.16438922228 + \frac{y}{x \cdot x}\right) - 110.1139242984811\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 2}{\sqrt{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}} \cdot \left(x \cdot \left(y + x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right)\right) + z\right)}{\sqrt{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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 2.0) 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)))