Average Error: 26.1 → 1.2
Time: 17.7s
Precision: binary64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3945332052864718 \cdot 10^{+52}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922228 + \left(\frac{y}{{x}^{3}} + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) - \left(101.7851458539211 \cdot \frac{1}{x} + 124074.40615218398 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{elif}\;x \leq 1.5691069027695154 \cdot 10^{+36}:\\ \;\;\;\;\begin{array}{l} t_0 := {x}^{3} \cdot 43.3400022514\\ t_1 := {x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left(t_0 + x \cdot 313.399215894\right)\right)\right)\\ t_2 := \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, t_0\right)\right)\\ \left(70.37071397084 \cdot \frac{{x}^{4}}{t_1} + \left(\frac{y \cdot {x}^{2}}{t_1} + \left(\frac{{x}^{5}}{\left(-47.066876606 - t_2\right) - {x}^{4}} \cdot -4.16438922228 + \frac{x \cdot z}{t_1}\right)\right)\right) - \left(2 \cdot \frac{x \cdot y}{t_1} + \left(2 \cdot \frac{z}{t_1} + \left(275.038832832 \cdot \left(x \cdot \frac{x}{{x}^{4} + \left(47.066876606 + t_2\right)}\right) + 19.8795684148 \cdot \frac{{x}^{3}}{t_1}\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\\ \end{array} \]
\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 -1.3945332052864718 \cdot 10^{+52}:\\
\;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922228 + \left(\frac{y}{{x}^{3}} + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) - \left(101.7851458539211 \cdot \frac{1}{x} + 124074.40615218398 \cdot \frac{1}{{x}^{3}}\right)\right)\\

\mathbf{elif}\;x \leq 1.5691069027695154 \cdot 10^{+36}:\\
\;\;\;\;\begin{array}{l}
t_0 := {x}^{3} \cdot 43.3400022514\\
t_1 := {x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left(t_0 + x \cdot 313.399215894\right)\right)\right)\\
t_2 := \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, t_0\right)\right)\\
\left(70.37071397084 \cdot \frac{{x}^{4}}{t_1} + \left(\frac{y \cdot {x}^{2}}{t_1} + \left(\frac{{x}^{5}}{\left(-47.066876606 - t_2\right) - {x}^{4}} \cdot -4.16438922228 + \frac{x \cdot z}{t_1}\right)\right)\right) - \left(2 \cdot \frac{x \cdot y}{t_1} + \left(2 \cdot \frac{z}{t_1} + \left(275.038832832 \cdot \left(x \cdot \frac{x}{{x}^{4} + \left(47.066876606 + t_2\right)}\right) + 19.8795684148 \cdot \frac{{x}^{3}}{t_1}\right)\right)\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\\


\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 (<= x -1.3945332052864718e+52)
   (*
    (- x 2.0)
    (-
     (+
      4.16438922228
      (+ (/ y (pow x 3.0)) (* 3451.550173699799 (/ 1.0 (pow x 2.0)))))
     (+
      (* 101.7851458539211 (/ 1.0 x))
      (* 124074.40615218398 (/ 1.0 (pow x 3.0))))))
   (if (<= x 1.5691069027695154e+36)
     (let* ((t_0 (* (pow x 3.0) 43.3400022514))
            (t_1
             (+
              (pow x 4.0)
              (+
               47.066876606
               (+ (* (pow x 2.0) 263.505074721) (+ t_0 (* x 313.399215894))))))
            (t_2 (fma (* x x) 263.505074721 (fma x 313.399215894 t_0))))
       (-
        (+
         (* 70.37071397084 (/ (pow x 4.0) t_1))
         (+
          (/ (* y (pow x 2.0)) t_1)
          (+
           (*
            (/ (pow x 5.0) (- (- -47.066876606 t_2) (pow x 4.0)))
            -4.16438922228)
           (/ (* x z) t_1))))
        (+
         (* 2.0 (/ (* x y) t_1))
         (+
          (* 2.0 (/ z t_1))
          (+
           (* 275.038832832 (* x (/ x (+ (pow x 4.0) (+ 47.066876606 t_2)))))
           (* 19.8795684148 (/ (pow x 3.0) t_1)))))))
     (-
      (+ (fma x 4.16438922228 (/ 3655.1204654076414 x)) (/ y (* x x)))
      (+ 110.1139242984811 (/ 130977.50649958357 (* x x)))))))
double code(double x, double y, double z) {
	return ((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);
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3945332052864718e+52) {
		tmp = (x - 2.0) * ((4.16438922228 + ((y / pow(x, 3.0)) + (3451.550173699799 * (1.0 / pow(x, 2.0))))) - ((101.7851458539211 * (1.0 / x)) + (124074.40615218398 * (1.0 / pow(x, 3.0)))));
	} else if (x <= 1.5691069027695154e+36) {
		double t_0 = pow(x, 3.0) * 43.3400022514;
		double t_1 = pow(x, 4.0) + (47.066876606 + ((pow(x, 2.0) * 263.505074721) + (t_0 + (x * 313.399215894))));
		double t_2 = fma((x * x), 263.505074721, fma(x, 313.399215894, t_0));
		tmp = ((70.37071397084 * (pow(x, 4.0) / t_1)) + (((y * pow(x, 2.0)) / t_1) + (((pow(x, 5.0) / ((-47.066876606 - t_2) - pow(x, 4.0))) * -4.16438922228) + ((x * z) / t_1)))) - ((2.0 * ((x * y) / t_1)) + ((2.0 * (z / t_1)) + ((275.038832832 * (x * (x / (pow(x, 4.0) + (47.066876606 + t_2))))) + (19.8795684148 * (pow(x, 3.0) / t_1)))));
	} else {
		tmp = (fma(x, 4.16438922228, (3655.1204654076414 / x)) + (y / (x * x))) - (110.1139242984811 + (130977.50649958357 / (x * x)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.1
Target0.8
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3945332052864718e52

    1. Initial program 62.4

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

    2. Simplified62.4

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

    3. Applied *-un-lft-identity_binary6462.4

      \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

    4. Applied times-frac_binary6459.0

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

    5. Simplified59.0

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

    6. Taylor expanded in x around inf 1.0

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(4.16438922228 + \left(\frac{y}{{x}^{3}} + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) - \left(101.7851458539211 \cdot \frac{1}{x} + 124074.40615218398 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

    if -1.3945332052864718e52 < x < 1.5691069027695154e36

    1. Initial program 1.1

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

    2. Simplified1.1

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

    3. Taylor expanded in y around 0 1.0

      \[\leadsto \color{blue}{\left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \frac{{x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right)} \]

    4. Applied *-un-lft-identity_binary641.0

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \frac{{x}^{2}}{\color{blue}{1 \cdot \left({x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)\right)}} + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    5. Applied add-sqr-sqrt_binary6432.8

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \frac{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2}}{1 \cdot \left({x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)\right)} + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    6. Applied unpow-prod-down_binary6432.8

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \frac{\color{blue}{{\left(\sqrt{x}\right)}^{2} \cdot {\left(\sqrt{x}\right)}^{2}}}{1 \cdot \left({x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)\right)} + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    7. Applied times-frac_binary6432.8

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \color{blue}{\left(\frac{{\left(\sqrt{x}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{x}\right)}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)} + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    8. Simplified32.8

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \left(\color{blue}{x} \cdot \frac{{\left(\sqrt{x}\right)}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right) + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    9. Simplified1.0

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \left(x \cdot \color{blue}{\frac{x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, 43.3400022514 \cdot {x}^{3}\right)\right)\right)}}\right) + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    10. Applied frac-2neg_binary641.0

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \color{blue}{\frac{-{x}^{5}}{-\left({x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)\right)}} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \left(x \cdot \frac{x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, 43.3400022514 \cdot {x}^{3}\right)\right)\right)}\right) + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    11. Simplified1.0

      \[\leadsto \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{-{x}^{5}}{\color{blue}{\left(-47.066876606 - \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, 43.3400022514 \cdot {x}^{3}\right)\right)\right) - {x}^{4}}} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \left(x \cdot \frac{x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, 43.3400022514 \cdot {x}^{3}\right)\right)\right)}\right) + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) \]

    if 1.5691069027695154e36 < x

    1. Initial program 59.4

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

    2. Simplified59.4

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

    3. Taylor expanded in x around inf 1.7

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + \left(3655.1204654076414 \cdot \frac{1}{x} + 4.16438922228 \cdot x\right)\right) - \left(110.1139242984811 + 130977.50649958357 \cdot \frac{1}{{x}^{2}}\right)} \]

    4. Simplified1.7

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3945332052864718 \cdot 10^{+52}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922228 + \left(\frac{y}{{x}^{3}} + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) - \left(101.7851458539211 \cdot \frac{1}{x} + 124074.40615218398 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{elif}\;x \leq 1.5691069027695154 \cdot 10^{+36}:\\ \;\;\;\;\left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left({x}^{3} \cdot 43.3400022514 + x \cdot 313.399215894\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left({x}^{3} \cdot 43.3400022514 + x \cdot 313.399215894\right)\right)\right)} + \left(\frac{{x}^{5}}{\left(-47.066876606 - \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right) - {x}^{4}} \cdot -4.16438922228 + \frac{x \cdot z}{{x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left({x}^{3} \cdot 43.3400022514 + x \cdot 313.399215894\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{x \cdot y}{{x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left({x}^{3} \cdot 43.3400022514 + x \cdot 313.399215894\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left({x}^{3} \cdot 43.3400022514 + x \cdot 313.399215894\right)\right)\right)} + \left(275.038832832 \cdot \left(x \cdot \frac{x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}\right) + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left({x}^{2} \cdot 263.505074721 + \left({x}^{3} \cdot 43.3400022514 + x \cdot 313.399215894\right)\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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)))