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

Average Error: 26.6 → 0.9

Time: 25.5s

Precision: binary64

Cost: 46536

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+76}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+60}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \left(-110.1139242984811 + \frac{y + -130977.50649958357}{x \cdot x}\right)\\ \end{array} \]

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 -7e+76)
   (/ (+ x -2.0) 0.24013125253755718)
   (if (<= x 2.2e+60)
     (*
      (+ 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)))
     (+
      (fma x 4.16438922228 (/ 3655.1204654076414 x))
      (+ -110.1139242984811 (/ (+ y -130977.50649958357) (* 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 <= -7e+76) {
		tmp = (x + -2.0) / 0.24013125253755718;
	} else if (x <= 2.2e+60) {
		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 = fma(x, 4.16438922228, (3655.1204654076414 / x)) + (-110.1139242984811 + ((y + -130977.50649958357) / (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 <= -7e+76)
		tmp = Float64(Float64(x + -2.0) / 0.24013125253755718);
	elseif (x <= 2.2e+60)
		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(fma(x, 4.16438922228, Float64(3655.1204654076414 / x)) + Float64(-110.1139242984811 + Float64(Float64(y + -130977.50649958357) / 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, -7e+76], N[(N[(x + -2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision], If[LessEqual[x, 2.2e+60], 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[(x * 4.16438922228 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] + N[(-110.1139242984811 + N[(N[(y + -130977.50649958357), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	26.6
Target	0.8
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if x < -7.00000000000000001e76

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      associate-/l* [=>] 64.0	\[ \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]
      sub-neg [=>] 64.0	\[ \frac{\color{blue}{x + \left(-2\right)}}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}} \]
      metadata-eval [=>] 64.0	\[ \frac{x + \color{blue}{-2}}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}} \]
      fma-def [=>] 64.0	\[ \frac{x + -2}{\frac{\color{blue}{\mathsf{fma}\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894, x, 47.066876606\right)}}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}} \]
      fma-def [=>] 64.0	\[ \frac{x + -2}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721, x, 313.399215894\right)}, x, 47.066876606\right)}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}} \]
      fma-def [=>] 64.0	\[ \frac{x + -2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right)}, x, 313.399215894\right), x, 47.066876606\right)}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}} \]
      fma-def [=>] 64.0	\[ \frac{x + -2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y, x, z\right)}}} \]
      fma-def [=>] 64.0	\[ \frac{x + -2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416, x, y\right)}, x, z\right)}} \]
      fma-def [=>] 64.0	\[ \frac{x + -2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot 4.16438922228 + 78.6994924154, x, 137.519416416\right)}, x, y\right), x, z\right)}} \]
      fma-def [=>] 64.0	\[ \frac{x + -2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right)}, x, 137.519416416\right), x, y\right), x, z\right)}} \]

   3. Taylor expanded in x around inf 0.8

      \[\leadsto \frac{x + -2}{\color{blue}{0.24013125253755718}} \]

   if -7.00000000000000001e76 < x < 2.19999999999999996e60

   1. Initial program 3.0

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

   2. Simplified 1.0

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

      [Start] 3.0	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      associate-*r/ [<=] 1.0	\[ \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \]
      sub-neg [=>] 1.0	\[ \color{blue}{\left(x + \left(-2\right)\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      metadata-eval [=>] 1.0	\[ \left(x + \color{blue}{-2}\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\color{blue}{x \cdot \left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right)} + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y, z\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right)} + y, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416, y\right)}, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)} + 137.519416416, y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot 4.16438922228 + 78.6994924154, 137.519416416\right)}, y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right)}, 137.519416416\right), y\right), z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 1.0	\[ \left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\color{blue}{x \cdot \left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right)} + 47.066876606} \]

   if 2.19999999999999996e60 < x

   1. Initial program 63.8

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

   2. Simplified 60.1

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

      [Start] 63.8	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      associate-*l/ [<=] 60.1	\[ \color{blue}{\frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)} \]
      *-commutative [=>] 60.1	\[ \color{blue}{\left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \]
      *-commutative [=>] 60.1	\[ \left(\color{blue}{x \cdot \left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right)} + z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 60.1	\[ \color{blue}{\mathsf{fma}\left(x, \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y, z\right)} \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 60.1	\[ \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right)} + y, z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 60.1	\[ \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416, y\right)}, z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      *-commutative [=>] 60.1	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)} + 137.519416416, y\right), z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 60.1	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot 4.16438922228 + 78.6994924154, 137.519416416\right)}, y\right), z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      fma-def [=>] 60.1	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right)}, 137.519416416\right), y\right), z\right) \cdot \frac{x - 2}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      sub-neg [=>] 60.1	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{\color{blue}{x + \left(-2\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      metadata-eval [=>] 60.1	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{x + \color{blue}{-2}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   3. Taylor expanded in x around -inf 0.7

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

   4. Simplified 0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) - \left(\frac{130977.50649958357 - y}{x \cdot x} - -110.1139242984811\right)} \]
      Proof

      [Start] 0.7	\[ \left(-1 \cdot \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} + \left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right)\right) - 110.1139242984811 \]
      sub-neg [=>] 0.7	\[ \color{blue}{\left(-1 \cdot \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} + \left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right)\right) + \left(-110.1139242984811\right)} \]
      +-commutative [=>] 0.7	\[ \color{blue}{\left(\left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right) + -1 \cdot \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}\right)} + \left(-110.1139242984811\right) \]
      mul-1-neg [=>] 0.7	\[ \left(\left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right) + \color{blue}{\left(-\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}\right)}\right) + \left(-110.1139242984811\right) \]
      unsub-neg [=>] 0.7	\[ \color{blue}{\left(\left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right) - \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}\right)} + \left(-110.1139242984811\right) \]
      associate-+l- [=>] 0.7	\[ \color{blue}{\left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right) - \left(\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} - \left(-110.1139242984811\right)\right)} \]
      *-commutative [=>] 0.7	\[ \left(\color{blue}{x \cdot 4.16438922228} + 3655.1204654076414 \cdot \frac{1}{x}\right) - \left(\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} - \left(-110.1139242984811\right)\right) \]
      fma-def [=>] 0.7	\[ \color{blue}{\mathsf{fma}\left(x, 4.16438922228, 3655.1204654076414 \cdot \frac{1}{x}\right)} - \left(\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} - \left(-110.1139242984811\right)\right) \]
      associate-*r/ [=>] 0.7	\[ \mathsf{fma}\left(x, 4.16438922228, \color{blue}{\frac{3655.1204654076414 \cdot 1}{x}}\right) - \left(\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} - \left(-110.1139242984811\right)\right) \]
      metadata-eval [=>] 0.7	\[ \mathsf{fma}\left(x, 4.16438922228, \frac{\color{blue}{3655.1204654076414}}{x}\right) - \left(\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} - \left(-110.1139242984811\right)\right) \]
      mul-1-neg [=>] 0.7	\[ \mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) - \left(\frac{130977.50649958357 + \color{blue}{\left(-y\right)}}{{x}^{2}} - \left(-110.1139242984811\right)\right) \]
      unsub-neg [=>] 0.7	\[ \mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) - \left(\frac{\color{blue}{130977.50649958357 - y}}{{x}^{2}} - \left(-110.1139242984811\right)\right) \]
      unpow2 [=>] 0.7	\[ \mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) - \left(\frac{130977.50649958357 - y}{\color{blue}{x \cdot x}} - \left(-110.1139242984811\right)\right) \]
      metadata-eval [=>] 0.7	\[ \mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) - \left(\frac{130977.50649958357 - y}{x \cdot x} - \color{blue}{-110.1139242984811}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+76}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+60}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \left(-110.1139242984811 + \frac{y + -130977.50649958357}{x \cdot x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	7624

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1500000000:\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(\frac{x}{t_1} + -2 \cdot \frac{1}{t_1}\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 - x \cdot -4.16438922228\right)\right)\right) \cdot \left(x \cdot \left(x + -2\right)\right)}{t_0} - z \cdot \left(-2 \cdot \frac{-1}{t_0} - \frac{x}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \left(-110.1139242984811 + \frac{y + -130977.50649958357}{x \cdot x}\right)\\ \end{array} \]

Alternative 2
Error	1.9
Cost	5064

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1500000000:\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(\frac{x}{t_1} + -2 \cdot \frac{1}{t_1}\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{\left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 - x \cdot -4.16438922228\right)\right)\right) \cdot \left(x \cdot \left(x + -2\right)\right)}{t_0} - z \cdot \left(-2 \cdot \frac{-1}{t_0} - \frac{x}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) + \frac{1}{x \cdot x} \cdot \frac{y + -124074.40615218398}{x}\right) + \frac{-101.7851458539211}{x}\right)\\ \end{array} \]

Alternative 3
Error	2.2
Cost	3913

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)\\ \mathbf{if}\;x \leq -1500000000 \lor \neg \left(x \leq 1.85 \cdot 10^{+30}\right):\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(\frac{x}{t_0} + -2 \cdot \frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 - x \cdot -4.16438922228\right)\right)\right) \cdot \left(x \cdot \left(x + -2\right)\right)}{t_1} + \frac{\left(x + -2\right) \cdot z}{t_1}\\ \end{array} \]

Alternative 4
Error	2.2
Cost	2761

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1500000000 \lor \neg \left(x \leq 2.4 \cdot 10^{+30}\right):\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(\frac{x}{t_0} + -2 \cdot \frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 - x \cdot -4.16438922228\right)\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	2632

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{+46}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+50}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 - x \cdot -4.16438922228\right)\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) + \frac{1}{x \cdot x} \cdot \frac{y + -124074.40615218398}{x}\right) + \frac{-101.7851458539211}{x}\right)\\ \end{array} \]

Alternative 6
Error	3.2
Cost	2120

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot 137.519416416\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) + \frac{1}{x \cdot x} \cdot \frac{y + -124074.40615218398}{x}\right) + \frac{-101.7851458539211}{x}\right)\\ \end{array} \]

Alternative 7
Error	5.7
Cost	1992

\[\begin{array}{l} \mathbf{if}\;x \leq -1550000000:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 16:\\ \;\;\;\;x \cdot \left(y \cdot -0.0424927283095952 - z \cdot -0.3041881842569256\right) + z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) + \frac{1}{x \cdot x} \cdot \frac{y + -124074.40615218398}{x}\right) + \frac{-101.7851458539211}{x}\right)\\ \end{array} \]

Alternative 8
Error	6.7
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x \leq -1550000000 \lor \neg \left(x \leq 6 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(z \cdot 0.0212463641547976 + x \cdot \left(y \cdot 0.0212463641547976 + z \cdot -0.14147091005106402\right)\right)\\ \end{array} \]

Alternative 9
Error	15.3
Cost	1100

\[\begin{array}{l} t_0 := \frac{x + -2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -1550000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+28}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{z}{47.066876606 + x \cdot 313.399215894}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	6.8
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -1550000000 \lor \neg \left(x \leq 1.9 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot -0.0424927283095952 - z \cdot -0.3041881842569256\right) + z \cdot -0.0424927283095952\\ \end{array} \]

Alternative 11
Error	15.4
Cost	844

\[\begin{array}{l} t_0 := \frac{x + -2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -1550000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+28}:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 - x \cdot -0.3041881842569256\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	15.3
Cost	716

\[\begin{array}{l} t_0 := \frac{x + -2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -1550000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-59}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+18}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	15.5
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -41000000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-59}:\\ \;\;\;\;-0.0424927283095952 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+18}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 14
Error	15.4
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -1550000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+18}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 15
Error	15.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1550000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+18}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 16
Error	62.6
Cost	192

\[x \cdot -0.3407596943375357 \]

Alternative 17
Error	35.3
Cost	192

\[x \cdot 4.16438922228 \]

Reproduce

herbie shell --seed 2023060 
(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)))