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

Average Error: 27.5 → 1.3

Time: 27.2s

Precision: binary64

Cost: 14660

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+49}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(4.16438922228 + \left(-\frac{124074.40615218398 + \left(-y\right)}{{x}^{3}}\right)\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \frac{\left(-y\right) + 130977.50649958357}{-{x}^{2}}\right) - 110.1139242984811\\ \end{array} \]

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 -2.6e+49)
   (*
    (- x 2.0)
    (-
     (+
      (* 3451.550173699799 (/ 1.0 (pow x 2.0)))
      (+ 4.16438922228 (- (/ (+ 124074.40615218398 (- y)) (pow x 3.0)))))
     (* 101.7851458539211 (/ 1.0 x))))
   (if (<= x 1.35e+16)
     (*
      (- x 2.0)
      (/
       (+
        (*
         x
         (+
          (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
          y))
        z)
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606)))
     (-
      (+
       (+ (* x 4.16438922228) (* (/ 1.0 x) 3655.1204654076414))
       (/ (+ (- y) 130977.50649958357) (- (pow x 2.0))))
      110.1139242984811))))

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 <= -2.6e+49) {
		tmp = (x - 2.0) * (((3451.550173699799 * (1.0 / pow(x, 2.0))) + (4.16438922228 + -((124074.40615218398 + -y) / pow(x, 3.0)))) - (101.7851458539211 * (1.0 / x)));
	} else if (x <= 1.35e+16) {
		tmp = (x - 2.0) * (((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606));
	} else {
		tmp = (((x * 4.16438922228) + ((1.0 / x) * 3655.1204654076414)) + ((-y + 130977.50649958357) / -pow(x, 2.0))) - 110.1139242984811;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - 2.0d0) * ((((((((x * 4.16438922228d0) + 78.6994924154d0) * x) + 137.519416416d0) * x) + y) * x) + z)) / (((((((x + 43.3400022514d0) * x) + 263.505074721d0) * x) + 313.399215894d0) * x) + 47.066876606d0)
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.6d+49)) then
        tmp = (x - 2.0d0) * (((3451.550173699799d0 * (1.0d0 / (x ** 2.0d0))) + (4.16438922228d0 + -((124074.40615218398d0 + -y) / (x ** 3.0d0)))) - (101.7851458539211d0 * (1.0d0 / x)))
    else if (x <= 1.35d+16) then
        tmp = (x - 2.0d0) * (((x * ((x * ((x * ((x * 4.16438922228d0) + 78.6994924154d0)) + 137.519416416d0)) + y)) + z) / ((x * ((x * ((x * (x + 43.3400022514d0)) + 263.505074721d0)) + 313.399215894d0)) + 47.066876606d0))
    else
        tmp = (((x * 4.16438922228d0) + ((1.0d0 / x) * 3655.1204654076414d0)) + ((-y + 130977.50649958357d0) / -(x ** 2.0d0))) - 110.1139242984811d0
    end if
    code = tmp
end function

Java

public static 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);
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.6e+49) {
		tmp = (x - 2.0) * (((3451.550173699799 * (1.0 / Math.pow(x, 2.0))) + (4.16438922228 + -((124074.40615218398 + -y) / Math.pow(x, 3.0)))) - (101.7851458539211 * (1.0 / x)));
	} else if (x <= 1.35e+16) {
		tmp = (x - 2.0) * (((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606));
	} else {
		tmp = (((x * 4.16438922228) + ((1.0 / x) * 3655.1204654076414)) + ((-y + 130977.50649958357) / -Math.pow(x, 2.0))) - 110.1139242984811;
	}
	return tmp;
}

Python

def code(x, y, 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)

↓

def code(x, y, z):
	tmp = 0
	if x <= -2.6e+49:
		tmp = (x - 2.0) * (((3451.550173699799 * (1.0 / math.pow(x, 2.0))) + (4.16438922228 + -((124074.40615218398 + -y) / math.pow(x, 3.0)))) - (101.7851458539211 * (1.0 / x)))
	elif x <= 1.35e+16:
		tmp = (x - 2.0) * (((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606))
	else:
		tmp = (((x * 4.16438922228) + ((1.0 / x) * 3655.1204654076414)) + ((-y + 130977.50649958357) / -math.pow(x, 2.0))) - 110.1139242984811
	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 <= -2.6e+49)
		tmp = Float64(Float64(x - 2.0) * Float64(Float64(Float64(3451.550173699799 * Float64(1.0 / (x ^ 2.0))) + Float64(4.16438922228 + Float64(-Float64(Float64(124074.40615218398 + Float64(-y)) / (x ^ 3.0))))) - Float64(101.7851458539211 * Float64(1.0 / x))));
	elseif (x <= 1.35e+16)
		tmp = Float64(Float64(x - 2.0) * Float64(Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)));
	else
		tmp = Float64(Float64(Float64(Float64(x * 4.16438922228) + Float64(Float64(1.0 / x) * 3655.1204654076414)) + Float64(Float64(Float64(-y) + 130977.50649958357) / Float64(-(x ^ 2.0)))) - 110.1139242984811);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((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);
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.6e+49)
		tmp = (x - 2.0) * (((3451.550173699799 * (1.0 / (x ^ 2.0))) + (4.16438922228 + -((124074.40615218398 + -y) / (x ^ 3.0)))) - (101.7851458539211 * (1.0 / x)));
	elseif (x <= 1.35e+16)
		tmp = (x - 2.0) * (((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606));
	else
		tmp = (((x * 4.16438922228) + ((1.0 / x) * 3655.1204654076414)) + ((-y + 130977.50649958357) / -(x ^ 2.0))) - 110.1139242984811;
	end
	tmp_2 = 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, -2.6e+49], N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(3451.550173699799 * N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.16438922228 + (-N[(N[(124074.40615218398 + (-y)), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[(101.7851458539211 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+16], N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * 3655.1204654076414), $MachinePrecision]), $MachinePrecision] + N[(N[((-y) + 130977.50649958357), $MachinePrecision] / (-N[Power[x, 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - 110.1139242984811), $MachinePrecision]]]

Target

Original	27.5
Target	0.8
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999989e49

   1. Initial program 61.6

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

   2. Simplified 57.6

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

      [Start] 61.6	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 61.6	\[ \frac{\color{blue}{\left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right) \cdot \left(x - 2\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-47 [=>] 57.6	\[ \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \]
      rational_best-simplify-2 [=>] 57.6	\[ \left(x - 2\right) \cdot \frac{\color{blue}{x \cdot \left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right)} + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 57.6	\[ \left(x - 2\right) \cdot \frac{x \cdot \left(\color{blue}{x \cdot \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right)} + y\right) + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 57.6	\[ \left(x - 2\right) \cdot \frac{x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)} + 137.519416416\right) + y\right) + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   3. Taylor expanded in x around -inf 1.4

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

   4. Simplified 1.4

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

      [Start] 1.4	\[ \left(x - 2\right) \cdot \left(\left(-1 \cdot \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}} + \left(4.16438922228 + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right) \]
      rational_best-simplify-43 [=>] 1.4	\[ \left(x - 2\right) \cdot \left(\color{blue}{\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(4.16438922228 + -1 \cdot \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right)\right)} - 101.7851458539211 \cdot \frac{1}{x}\right) \]
      rational_best-simplify-2 [=>] 1.4	\[ \left(x - 2\right) \cdot \left(\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(4.16438922228 + \color{blue}{\frac{124074.40615218398 + -1 \cdot y}{{x}^{3}} \cdot -1}\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right) \]
      rational_best-simplify-12 [=>] 1.4	\[ \left(x - 2\right) \cdot \left(\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(4.16438922228 + \color{blue}{\left(-\frac{124074.40615218398 + -1 \cdot y}{{x}^{3}}\right)}\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right) \]
      rational_best-simplify-2 [=>] 1.4	\[ \left(x - 2\right) \cdot \left(\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(4.16438922228 + \left(-\frac{124074.40615218398 + \color{blue}{y \cdot -1}}{{x}^{3}}\right)\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right) \]
      rational_best-simplify-12 [=>] 1.4	\[ \left(x - 2\right) \cdot \left(\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(4.16438922228 + \left(-\frac{124074.40615218398 + \color{blue}{\left(-y\right)}}{{x}^{3}}\right)\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right) \]

   if -2.59999999999999989e49 < x < 1.35e16

   1. Initial program 1.0

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

   2. Simplified 0.5

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

      [Start] 1.0	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 1.0	\[ \frac{\color{blue}{\left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right) \cdot \left(x - 2\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-47 [=>] 0.5	\[ \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \]
      rational_best-simplify-2 [=>] 0.5	\[ \left(x - 2\right) \cdot \frac{\color{blue}{x \cdot \left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right)} + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 0.5	\[ \left(x - 2\right) \cdot \frac{x \cdot \left(\color{blue}{x \cdot \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right)} + y\right) + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 0.5	\[ \left(x - 2\right) \cdot \frac{x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)} + 137.519416416\right) + y\right) + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   if 1.35e16 < x

   1. Initial program 56.1

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

   2. Simplified 52.0

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

      [Start] 56.1	\[ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 56.1	\[ \frac{\color{blue}{\left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right) \cdot \left(x - 2\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-47 [=>] 52.0	\[ \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}} \]
      rational_best-simplify-2 [=>] 52.0	\[ \left(x - 2\right) \cdot \frac{\color{blue}{x \cdot \left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right)} + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 52.0	\[ \left(x - 2\right) \cdot \frac{x \cdot \left(\color{blue}{x \cdot \left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right)} + y\right) + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
      rational_best-simplify-2 [=>] 52.0	\[ \left(x - 2\right) \cdot \frac{x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)} + 137.519416416\right) + y\right) + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   3. Taylor expanded in x around -inf 2.8

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

   4. Simplified 2.8

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

      [Start] 2.8	\[ \left(-1 \cdot \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} + \left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right)\right) - 110.1139242984811 \]
      rational_best-simplify-1 [=>] 2.8	\[ \color{blue}{\left(\left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right) + -1 \cdot \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}\right)} - 110.1139242984811 \]
      rational_best-simplify-2 [=>] 2.8	\[ \left(\left(\color{blue}{x \cdot 4.16438922228} + 3655.1204654076414 \cdot \frac{1}{x}\right) + -1 \cdot \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}\right) - 110.1139242984811 \]
      rational_best-simplify-2 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \color{blue}{\frac{1}{x} \cdot 3655.1204654076414}\right) + -1 \cdot \frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}\right) - 110.1139242984811 \]
      rational_best-simplify-2 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \color{blue}{\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}} \cdot -1}\right) - 110.1139242984811 \]
      rational_best-simplify-12 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \color{blue}{\left(-\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}\right)}\right) - 110.1139242984811 \]
      rational_best-simplify-9 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \color{blue}{\frac{\frac{130977.50649958357 + -1 \cdot y}{{x}^{2}}}{-1}}\right) - 110.1139242984811 \]
      rational_best-simplify-45 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \color{blue}{\frac{\frac{130977.50649958357 + -1 \cdot y}{-1}}{{x}^{2}}}\right) - 110.1139242984811 \]
      rational_best-simplify-48 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \color{blue}{\frac{130977.50649958357 + -1 \cdot y}{{x}^{2} \cdot -1}}\right) - 110.1139242984811 \]
      rational_best-simplify-1 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \frac{\color{blue}{-1 \cdot y + 130977.50649958357}}{{x}^{2} \cdot -1}\right) - 110.1139242984811 \]
      rational_best-simplify-2 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \frac{\color{blue}{y \cdot -1} + 130977.50649958357}{{x}^{2} \cdot -1}\right) - 110.1139242984811 \]
      rational_best-simplify-12 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \frac{\color{blue}{\left(-y\right)} + 130977.50649958357}{{x}^{2} \cdot -1}\right) - 110.1139242984811 \]
      rational_best-simplify-12 [=>] 2.8	\[ \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \frac{\left(-y\right) + 130977.50649958357}{\color{blue}{-{x}^{2}}}\right) - 110.1139242984811 \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+49}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(4.16438922228 + \left(-\frac{124074.40615218398 + \left(-y\right)}{{x}^{3}}\right)\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \frac{\left(-y\right) + 130977.50649958357}{-{x}^{2}}\right) - 110.1139242984811\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	7944

\[\begin{array}{l} t_0 := \left(\left(x \cdot 4.16438922228 + \frac{1}{x} \cdot 3655.1204654076414\right) + \frac{\left(-y\right) + 130977.50649958357}{-{x}^{2}}\right) - 110.1139242984811\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.4
Cost	2632

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+54}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+60}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right) \cdot \frac{x - 2}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 3
Error	1.4
Cost	2632

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+54}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+55}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 4
Error	2.8
Cost	2504

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+54}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -190000000000:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \cdot x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(x \cdot 137.519416416 + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 5
Error	3.6
Cost	2120

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+21}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+30}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(x \cdot 137.519416416 + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 6
Error	5.0
Cost	1736

\[\begin{array}{l} \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;\left(x - 2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right) \cdot \left(x \cdot 0.3041881842569256 - 0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 - 110.1139242984811\\ \end{array} \]

Alternative 7
Error	5.0
Cost	1480

\[\begin{array}{l} \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;\left(x - 2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot 78.6994924154 + 137.519416416\right) + y\right) + z\right) \cdot \left(x \cdot 0.3041881842569256 - 0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 - 110.1139242984811\\ \end{array} \]

Alternative 8
Error	5.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;\left(x - 2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot \left(137.519416416 \cdot x + y\right) + z\right) \cdot \left(x \cdot 0.3041881842569256 - 0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 - 110.1139242984811\\ \end{array} \]

Alternative 9
Error	6.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;\left(x - 2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(0.0212463641547976 \cdot z + 0.0212463641547976 \cdot \left(y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 - 110.1139242984811\\ \end{array} \]

Alternative 10
Error	6.7
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;\left(x - 2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\left(-0.0424927283095952 \cdot y + 0.3041881842569256 \cdot z\right) \cdot x + -0.0424927283095952 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 - 110.1139242984811\\ \end{array} \]

Alternative 11
Error	15.6
Cost	848

\[\begin{array}{l} t_0 := -0.0424927283095952 \cdot \left(y \cdot x\right)\\ \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 11500000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 12
Error	15.6
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(-0.0424927283095952 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 11500000:\\ \;\;\;\;-0.0424927283095952 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 13
Error	15.6
Cost	848

\[\begin{array}{l} t_0 := \left(x - 2\right) \cdot 4.16438922228\\ \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(-0.0424927283095952 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 11500000:\\ \;\;\;\;-0.0424927283095952 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	15.5
Cost	848

\[\begin{array}{l} t_0 := x \cdot 4.16438922228 - 110.1139242984811\\ \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \left(-0.0424927283095952 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 11500000:\\ \;\;\;\;-0.0424927283095952 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	6.7
Cost	712

\[\begin{array}{l} t_0 := x \cdot 4.16438922228 - 110.1139242984811\\ \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \left(z + y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	6.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -76000000000:\\ \;\;\;\;\left(x - 2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;-0.0424927283095952 \cdot \left(z + y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 - 110.1139242984811\\ \end{array} \]

Alternative 17
Error	14.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-10}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 18
Error	34.6
Cost	192

\[x \cdot 4.16438922228 \]

Alternative 19
Error	61.9
Cost	64

\[-110.1139242984811 \]

Reproduce

herbie shell --seed 2023096 
(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)))