Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Accuracy: 57.8% → 98.0%

Time: 25.9s

Precision: binary64

Cost: 46536

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+64}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) + \frac{y + -124074.40615218398}{{x}^{3}}\right) + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+17}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \end{array} \]

FPCore

(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 -4.5e+64)
   (*
    (+ x -2.0)
    (+
     (+
      (+ 4.16438922228 (/ (/ 3451.550173699799 x) x))
      (/ (+ y -124074.40615218398) (pow x 3.0)))
     (/ -101.7851458539211 x)))
   (if (<= x 5.2e+17)
     (*
      (+ x -2.0)
      (/
       (fma
        x
        (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
        z)
       (fma
        x
        (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
        47.066876606)))
     (+
      (/ y (* x x))
      (+
       (- (/ 7085.836212289914 x) (* x -4.16438922228))
       (+ -188.81341671388108 (/ -258651.98023111798 (* x x))))))))

C

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 <= -4.5e+64) {
		tmp = (x + -2.0) * (((4.16438922228 + ((3451.550173699799 / x) / x)) + ((y + -124074.40615218398) / pow(x, 3.0))) + (-101.7851458539211 / x));
	} else if (x <= 5.2e+17) {
		tmp = (x + -2.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = (y / (x * x)) + (((7085.836212289914 / x) - (x * -4.16438922228)) + (-188.81341671388108 + (-258651.98023111798 / (x * x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.5e+64)
		tmp = Float64(Float64(x + -2.0) * Float64(Float64(Float64(4.16438922228 + Float64(Float64(3451.550173699799 / x) / x)) + Float64(Float64(y + -124074.40615218398) / (x ^ 3.0))) + Float64(-101.7851458539211 / x)));
	elseif (x <= 5.2e+17)
		tmp = Float64(Float64(x + -2.0) * Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = Float64(Float64(y / Float64(x * x)) + Float64(Float64(Float64(7085.836212289914 / x) - Float64(x * -4.16438922228)) + Float64(-188.81341671388108 + Float64(-258651.98023111798 / Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(x + 43.3400022514), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[x, -4.5e+64], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(N[(4.16438922228 + N[(N[(3451.550173699799 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(y + -124074.40615218398), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-101.7851458539211 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+17], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(7085.836212289914 / x), $MachinePrecision] - N[(x * -4.16438922228), $MachinePrecision]), $MachinePrecision] + N[(-188.81341671388108 + N[(-258651.98023111798 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	57.8%
Target	98.6%
Herbie	98.0%

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

Derivation

1. Split input into 3 regimes

2. if x < -4.49999999999999973e64

   1. Initial program 0.0%

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

   2. Simplified 3.6%

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

      [Start] 0.0	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      associate-*r/ [<=] 3.6	\[ \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \]
      sub-neg [=>] 3.6	\[ \color{blue}{\left(x + \left(-2\right)\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      metadata-eval [=>] 3.6	\[ \left(x + \color{blue}{-2}\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\color{blue}{x \cdot \left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right)} + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y, z\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right)} + y, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416, y\right)}, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)} + 137.519416416, y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot 4.16438922228 + 78.6994924154, 137.519416416\right)}, y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right)}, 137.519416416\right), y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 3.6	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\color{blue}{x \cdot \left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right)} + 47.066876606} \]

   3. Taylor expanded in x around -inf 98.8%

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

   4. Simplified 98.8%

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

      [Start] 98.8	\[ \left(x + -2\right) \cdot \left(\left(-1 \cdot \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}} + \left(4.16438922228 + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right) \]
      cancel-sign-sub-inv [=>] 98.8	\[ \left(x + -2\right) \cdot \color{blue}{\left(\left(-1 \cdot \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}} + \left(4.16438922228 + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right)} \]
      +-commutative [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\color{blue}{\left(\left(4.16438922228 + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right) + -1 \cdot \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right)} + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      mul-1-neg [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{\left(-\frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right)}\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      unsub-neg [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\color{blue}{\left(\left(4.16438922228 + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right) - \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right)} + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      associate-*r/ [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \color{blue}{\frac{3451.550173699799 \cdot 1}{{x}^{2}}}\right) - \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      metadata-eval [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\color{blue}{3451.550173699799}}{{x}^{2}}\right) - \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      unpow2 [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{3451.550173699799}{\color{blue}{x \cdot x}}\right) - \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      associate-/r* [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \color{blue}{\frac{\frac{3451.550173699799}{x}}{x}}\right) - \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      mul-1-neg [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) - \frac{124074.40615218398 + \color{blue}{\left(-y\right)}}{{x}^{3}}\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      unsub-neg [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) - \frac{\color{blue}{124074.40615218398 - y}}{{x}^{3}}\right) + \left(-101.7851458539211\right) \cdot \frac{1}{x}\right) \]
      associate-*r/ [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) - \frac{124074.40615218398 - y}{{x}^{3}}\right) + \color{blue}{\frac{\left(-101.7851458539211\right) \cdot 1}{x}}\right) \]
      metadata-eval [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) - \frac{124074.40615218398 - y}{{x}^{3}}\right) + \frac{\color{blue}{-101.7851458539211} \cdot 1}{x}\right) \]
      metadata-eval [=>] 98.8	\[ \left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) - \frac{124074.40615218398 - y}{{x}^{3}}\right) + \frac{\color{blue}{-101.7851458539211}}{x}\right) \]

   if -4.49999999999999973e64 < x < 5.2e17

   1. Initial program 97.4%

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

   2. Simplified 98.7%

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

      [Start] 97.4	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      associate-*r/ [<=] 98.7	\[ \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \]
      sub-neg [=>] 98.7	\[ \color{blue}{\left(x + \left(-2\right)\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      metadata-eval [=>] 98.7	\[ \left(x + \color{blue}{-2}\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\color{blue}{x \cdot \left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right)} + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y, z\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right)} + y, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416, y\right)}, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)} + 137.519416416, y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot 4.16438922228 + 78.6994924154, 137.519416416\right)}, y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right)}, 137.519416416\right), y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 98.7	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\color{blue}{x \cdot \left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right)} + 47.066876606} \]

   if 5.2e17 < x

   1. Initial program 11.8%

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

   2. Taylor expanded in x around inf 11.8%

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

   3. Simplified 11.8%

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

      [Start] 11.8	\[ \frac{\left(x - 2\right) \cdot \left(\left(4.16438922228 \cdot {x}^{3} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 11.8	\[ \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{{x}^{3} \cdot 4.16438922228} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   4. Taylor expanded in x around inf 95.6%

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + \left(7085.836212289914 \cdot \frac{1}{x} + 4.16438922228 \cdot x\right)\right) - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right)} \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x} + \left(\left(x \cdot 4.16438922228 + \frac{7085.836212289914}{x}\right) - \left(188.81341671388108 + \frac{258651.98023111798}{x \cdot x}\right)\right)} \]
      Proof

      [Start] 95.6	\[ \left(\frac{y}{{x}^{2}} + \left(7085.836212289914 \cdot \frac{1}{x} + 4.16438922228 \cdot x\right)\right) - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right) \]
      associate--l+ [=>] 95.6	\[ \color{blue}{\frac{y}{{x}^{2}} + \left(\left(7085.836212289914 \cdot \frac{1}{x} + 4.16438922228 \cdot x\right) - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right)\right)} \]
      unpow2 [=>] 95.6	\[ \frac{y}{\color{blue}{x \cdot x}} + \left(\left(7085.836212289914 \cdot \frac{1}{x} + 4.16438922228 \cdot x\right) - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right)\right) \]
      +-commutative [=>] 95.6	\[ \frac{y}{x \cdot x} + \left(\color{blue}{\left(4.16438922228 \cdot x + 7085.836212289914 \cdot \frac{1}{x}\right)} - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right)\right) \]
      *-commutative [=>] 95.6	\[ \frac{y}{x \cdot x} + \left(\left(\color{blue}{x \cdot 4.16438922228} + 7085.836212289914 \cdot \frac{1}{x}\right) - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right)\right) \]
      associate-*r/ [=>] 95.6	\[ \frac{y}{x \cdot x} + \left(\left(x \cdot 4.16438922228 + \color{blue}{\frac{7085.836212289914 \cdot 1}{x}}\right) - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right)\right) \]
      metadata-eval [=>] 95.6	\[ \frac{y}{x \cdot x} + \left(\left(x \cdot 4.16438922228 + \frac{\color{blue}{7085.836212289914}}{x}\right) - \left(188.81341671388108 + 258651.98023111798 \cdot \frac{1}{{x}^{2}}\right)\right) \]
      associate-*r/ [=>] 95.6	\[ \frac{y}{x \cdot x} + \left(\left(x \cdot 4.16438922228 + \frac{7085.836212289914}{x}\right) - \left(188.81341671388108 + \color{blue}{\frac{258651.98023111798 \cdot 1}{{x}^{2}}}\right)\right) \]
      metadata-eval [=>] 95.6	\[ \frac{y}{x \cdot x} + \left(\left(x \cdot 4.16438922228 + \frac{7085.836212289914}{x}\right) - \left(188.81341671388108 + \frac{\color{blue}{258651.98023111798}}{{x}^{2}}\right)\right) \]
      unpow2 [=>] 95.6	\[ \frac{y}{x \cdot x} + \left(\left(x \cdot 4.16438922228 + \frac{7085.836212289914}{x}\right) - \left(188.81341671388108 + \frac{258651.98023111798}{\color{blue}{x \cdot x}}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+64}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) + \frac{y + -124074.40615218398}{{x}^{3}}\right) + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+17}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.0%
Cost	7940

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+63}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) + \frac{y + -124074.40615218398}{{x}^{3}}\right) + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+18}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\frac{z}{t_0} - \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot -4.16438922228 + -78.6994924154\right) + -137.519416416\right) - y\right)}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	98.0%
Cost	3657

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+62} \lor \neg \left(x \leq 1.55 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\frac{z}{t_0} - \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot -4.16438922228 + -78.6994924154\right) + -137.519416416\right) - y\right)}{t_0}\right)\\ \end{array} \]

Alternative 3
Accuracy	97.8%
Cost	2633

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+29} \lor \neg \left(x \leq 1.12 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	96.4%
Cost	2121

\[\begin{array}{l} \mathbf{if}\;x \leq -7200000000000 \lor \neg \left(x \leq 2000\right):\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot 137.519416416\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	93.4%
Cost	1993

\[\begin{array}{l} \mathbf{if}\;x \leq -7200000000000 \lor \neg \left(x \leq 1850\right):\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\frac{z}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)} + 0.0212463641547976 \cdot \left(x \cdot y\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	92.9%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;x \leq -0.22 \lor \neg \left(x \leq 220\right):\\ \;\;\;\;\frac{y}{x \cdot x} + \left(\left(\frac{7085.836212289914}{x} - x \cdot -4.16438922228\right) + \left(-188.81341671388108 + \frac{-258651.98023111798}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0212463641547976 \cdot \left(z + -2 \cdot y\right) + z \cdot 0.28294182010212804\right) + z \cdot -0.0424927283095952\\ \end{array} \]

Alternative 7
Accuracy	90.8%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;x \leq -0.096:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{z}{47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)}\right)\\ \mathbf{elif}\;x \leq 230:\\ \;\;\;\;x \cdot \left(0.0212463641547976 \cdot \left(z + -2 \cdot y\right) + z \cdot 0.28294182010212804\right) + z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} + \frac{-55.572073733743466}{x \cdot x}\right)}\\ \end{array} \]

Alternative 8
Accuracy	90.3%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x \leq -0.18 \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} + \frac{-55.572073733743466}{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(z \cdot 0.0212463641547976 + x \cdot \left(y \cdot 0.0212463641547976 + z \cdot -0.14147091005106402\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	90.2%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x \leq -0.18 \lor \neg \left(x \leq 220\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} + \frac{-55.572073733743466}{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0212463641547976 \cdot \left(z + -2 \cdot y\right) + z \cdot 0.28294182010212804\right) + z \cdot -0.0424927283095952\\ \end{array} \]

Alternative 10
Accuracy	77.5%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \leq -9600 \lor \neg \left(x \leq 220\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \frac{5.86923874282773}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{47.066876606 + x \cdot \left(313.399215894 + x \cdot 263.505074721\right)}\\ \end{array} \]

Alternative 11
Accuracy	77.5%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \leq -31000 \lor \neg \left(x \leq 260\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} + \frac{-55.572073733743466}{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{47.066876606 + x \cdot \left(313.399215894 + x \cdot 263.505074721\right)}\\ \end{array} \]

Alternative 12
Accuracy	77.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -36 \lor \neg \left(x \leq 300\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \frac{5.86923874282773}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{47.066876606 + x \cdot 313.399215894}\\ \end{array} \]

Alternative 13
Accuracy	77.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5} \lor \neg \left(x \leq 420\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \frac{5.86923874282773}{x}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 + x \cdot 0.3041881842569256\right)\\ \end{array} \]

Alternative 14
Accuracy	76.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 4.16438922228 + -110.1139242984811\\ \mathbf{elif}\;x \leq 220:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 + x \cdot 0.3041881842569256\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \end{array} \]

Alternative 15
Accuracy	76.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 230:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 + x \cdot 0.3041881842569256\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \end{array} \]

Alternative 16
Accuracy	77.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{3655.1204654076414}{x} - x \cdot -4.16438922228\right) + -110.1139242984811\\ \mathbf{elif}\;x \leq 235:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 + x \cdot 0.3041881842569256\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \end{array} \]

Alternative 17
Accuracy	76.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5} \lor \neg \left(x \leq 9 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot 4.16438922228 + -110.1139242984811\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \end{array} \]

Alternative 18
Accuracy	76.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 4.16438922228 + -110.1139242984811\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-15}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \end{array} \]

Alternative 19
Accuracy	76.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 20
Accuracy	45.5%
Cost	192

\[x \cdot 4.16438922228 \]

Reproduce

herbie shell --seed 2023133 
(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)))