Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Accuracy: 53.2% → 81.6%

Time: 41.7s

Precision: binary64

Cost: 3404

Math

\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

↓

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ t_2 := \frac{t}{t_1}\\ t_3 := t_2 + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} + \left(\frac{a \cdot x - z}{\frac{y}{\frac{a}{y}}} - \frac{a}{\frac{y}{x}}\right)\right)\right)\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+156}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;t_2 + \frac{y}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{x \cdot x}\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+59}:\\ \;\;\;\;\frac{t + y \cdot \left(y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right) + 230661.510616\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ i (* y (+ c (* y (+ (* y (+ y a)) b))))))
        (t_2 (/ t t_1))
        (t_3
         (+
          t_2
          (+
           (+ x (/ 27464.7644705 (* y y)))
           (+ (/ z y) (- (/ (- (* a x) z) (/ y (/ a y))) (/ a (/ y x))))))))
   (if (<= y -1.3e+156)
     t_3
     (if (<= y -8.5e+21)
       (+ t_2 (/ y (+ (/ y x) (- (/ a x) (/ z (* x x))))))
       (if (<= y 6e+59)
         (/
          (+
           t
           (* y (+ (* y (+ 27464.7644705 (* y (+ z (* y x))))) 230661.510616)))
          t_1)
         t_3)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i + (y * (c + (y * ((y * (y + a)) + b))));
	double t_2 = t / t_1;
	double t_3 = t_2 + ((x + (27464.7644705 / (y * y))) + ((z / y) + ((((a * x) - z) / (y / (a / y))) - (a / (y / x)))));
	double tmp;
	if (y <= -1.3e+156) {
		tmp = t_3;
	} else if (y <= -8.5e+21) {
		tmp = t_2 + (y / ((y / x) + ((a / x) - (z / (x * x)))));
	} else if (y <= 6e+59) {
		tmp = (t + (y * ((y * (27464.7644705 + (y * (z + (y * x))))) + 230661.510616))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

↓

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = i + (y * (c + (y * ((y * (y + a)) + b))))
    t_2 = t / t_1
    t_3 = t_2 + ((x + (27464.7644705d0 / (y * y))) + ((z / y) + ((((a * x) - z) / (y / (a / y))) - (a / (y / x)))))
    if (y <= (-1.3d+156)) then
        tmp = t_3
    else if (y <= (-8.5d+21)) then
        tmp = t_2 + (y / ((y / x) + ((a / x) - (z / (x * x)))))
    else if (y <= 6d+59) then
        tmp = (t + (y * ((y * (27464.7644705d0 + (y * (z + (y * x))))) + 230661.510616d0))) / t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = i + (y * (c + (y * ((y * (y + a)) + b))));
	double t_2 = t / t_1;
	double t_3 = t_2 + ((x + (27464.7644705 / (y * y))) + ((z / y) + ((((a * x) - z) / (y / (a / y))) - (a / (y / x)))));
	double tmp;
	if (y <= -1.3e+156) {
		tmp = t_3;
	} else if (y <= -8.5e+21) {
		tmp = t_2 + (y / ((y / x) + ((a / x) - (z / (x * x)))));
	} else if (y <= 6e+59) {
		tmp = (t + (y * ((y * (27464.7644705 + (y * (z + (y * x))))) + 230661.510616))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

↓

def code(x, y, z, t, a, b, c, i):
	t_1 = i + (y * (c + (y * ((y * (y + a)) + b))))
	t_2 = t / t_1
	t_3 = t_2 + ((x + (27464.7644705 / (y * y))) + ((z / y) + ((((a * x) - z) / (y / (a / y))) - (a / (y / x)))))
	tmp = 0
	if y <= -1.3e+156:
		tmp = t_3
	elif y <= -8.5e+21:
		tmp = t_2 + (y / ((y / x) + ((a / x) - (z / (x * x)))))
	elif y <= 6e+59:
		tmp = (t + (y * ((y * (27464.7644705 + (y * (z + (y * x))))) + 230661.510616))) / t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

↓

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(i + Float64(y * Float64(c + Float64(y * Float64(Float64(y * Float64(y + a)) + b)))))
	t_2 = Float64(t / t_1)
	t_3 = Float64(t_2 + Float64(Float64(x + Float64(27464.7644705 / Float64(y * y))) + Float64(Float64(z / y) + Float64(Float64(Float64(Float64(a * x) - z) / Float64(y / Float64(a / y))) - Float64(a / Float64(y / x))))))
	tmp = 0.0
	if (y <= -1.3e+156)
		tmp = t_3;
	elseif (y <= -8.5e+21)
		tmp = Float64(t_2 + Float64(y / Float64(Float64(y / x) + Float64(Float64(a / x) - Float64(z / Float64(x * x))))));
	elseif (y <= 6e+59)
		tmp = Float64(Float64(t + Float64(y * Float64(Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(y * x))))) + 230661.510616))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

↓

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = i + (y * (c + (y * ((y * (y + a)) + b))));
	t_2 = t / t_1;
	t_3 = t_2 + ((x + (27464.7644705 / (y * y))) + ((z / y) + ((((a * x) - z) / (y / (a / y))) - (a / (y / x)))));
	tmp = 0.0;
	if (y <= -1.3e+156)
		tmp = t_3;
	elseif (y <= -8.5e+21)
		tmp = t_2 + (y / ((y / x) + ((a / x) - (z / (x * x)))));
	elseif (y <= 6e+59)
		tmp = (t + (y * ((y * (27464.7644705 + (y * (z + (y * x))))) + 230661.510616))) / t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i + N[(y * N[(c + N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[(x + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / y), $MachinePrecision] + N[(N[(N[(N[(a * x), $MachinePrecision] - z), $MachinePrecision] / N[(y / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3e+156], t$95$3, If[LessEqual[y, -8.5e+21], N[(t$95$2 + N[(y / N[(N[(y / x), $MachinePrecision] + N[(N[(a / x), $MachinePrecision] - N[(z / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+59], N[(N[(t + N[(y * N[(N[(y * N[(27464.7644705 + N[(y * N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000009e156 or 6.0000000000000001e59 < y

   1. Initial program 1.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in t around inf 1.0%

      \[\leadsto \color{blue}{\frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}} \]

   3. Taylor expanded in b around 0 1.0%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y}} \]

   4. Simplified 1.7%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\frac{y}{\frac{i + \left(y \cdot c + {y}^{3} \cdot \left(y + a\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \mathsf{fma}\left(y, x, z\right)\right)}}} \]
      Proof

      [Start] 1.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y} \]
      *-commutative [=>] 1.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{\color{blue}{y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right)}}{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y} \]
      associate-/l* [=>] 1.8	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\frac{y}{\frac{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}}} \]
      *-commutative [=>] 1.8	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \color{blue}{y \cdot \left(c + \left(y + a\right) \cdot {y}^{2}\right)}}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      *-commutative [=>] 1.8	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + y \cdot \left(c + \color{blue}{{y}^{2} \cdot \left(y + a\right)}\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      +-commutative [=>] 1.8	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + y \cdot \left(c + {y}^{2} \cdot \color{blue}{\left(a + y\right)}\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      distribute-lft-in [=>] 1.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \color{blue}{\left(y \cdot c + y \cdot \left({y}^{2} \cdot \left(a + y\right)\right)\right)}}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      +-commutative [<=] 1.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + y \cdot \left({y}^{2} \cdot \color{blue}{\left(y + a\right)}\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      associate-*r* [=>] 1.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + \color{blue}{\left(y \cdot {y}^{2}\right) \cdot \left(y + a\right)}\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      unpow2 [=>] 1.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(y + a\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      cube-mult [<=] 1.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + \color{blue}{{y}^{3}} \cdot \left(y + a\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]

   5. Taylor expanded in y around inf 65.7%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\left(\left(\frac{z}{y} + \left(27464.7644705 \cdot \frac{1}{{y}^{2}} + x\right)\right) - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right)} \]

   6. Simplified 73.7%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} - \left(\frac{a}{\frac{y}{x}} + \frac{z - a \cdot x}{\frac{y}{\frac{a}{y}}}\right)\right)\right)} \]
      Proof

      [Start] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(\frac{z}{y} + \left(27464.7644705 \cdot \frac{1}{{y}^{2}} + x\right)\right) - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right) \]
      +-commutative [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\color{blue}{\left(\left(27464.7644705 \cdot \frac{1}{{y}^{2}} + x\right) + \frac{z}{y}\right)} - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right) \]
      associate--l+ [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\left(\left(27464.7644705 \cdot \frac{1}{{y}^{2}} + x\right) + \left(\frac{z}{y} - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right)\right)} \]
      +-commutative [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\color{blue}{\left(x + 27464.7644705 \cdot \frac{1}{{y}^{2}}\right)} + \left(\frac{z}{y} - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right)\right) \]
      associate-*r/ [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \color{blue}{\frac{27464.7644705 \cdot 1}{{y}^{2}}}\right) + \left(\frac{z}{y} - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right)\right) \]
      metadata-eval [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \frac{\color{blue}{27464.7644705}}{{y}^{2}}\right) + \left(\frac{z}{y} - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right)\right) \]
      unpow2 [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \frac{27464.7644705}{\color{blue}{y \cdot y}}\right) + \left(\frac{z}{y} - \left(\frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}} + \frac{a \cdot x}{y}\right)\right)\right) \]
      +-commutative [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} - \color{blue}{\left(\frac{a \cdot x}{y} + \frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}}\right)}\right)\right) \]
      associate-/l* [=>] 65.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} - \left(\color{blue}{\frac{a}{\frac{y}{x}}} + \frac{\left(z - a \cdot x\right) \cdot a}{{y}^{2}}\right)\right)\right) \]
      associate-/l* [=>] 73.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} - \left(\frac{a}{\frac{y}{x}} + \color{blue}{\frac{z - a \cdot x}{\frac{{y}^{2}}{a}}}\right)\right)\right) \]
      unpow2 [=>] 73.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} - \left(\frac{a}{\frac{y}{x}} + \frac{z - a \cdot x}{\frac{\color{blue}{y \cdot y}}{a}}\right)\right)\right) \]
      associate-/l* [=>] 73.7	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} - \left(\frac{a}{\frac{y}{x}} + \frac{z - a \cdot x}{\color{blue}{\frac{y}{\frac{a}{y}}}}\right)\right)\right) \]

   if -1.30000000000000009e156 < y < -8.5e21

   1. Initial program 14.9%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in t around inf 14.9%

      \[\leadsto \color{blue}{\frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}} \]

   3. Taylor expanded in b around 0 12.8%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y}} \]

   4. Simplified 15.0%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\frac{y}{\frac{i + \left(y \cdot c + {y}^{3} \cdot \left(y + a\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \mathsf{fma}\left(y, x, z\right)\right)}}} \]
      Proof

      [Start] 12.8	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y} \]
      *-commutative [=>] 12.8	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{\color{blue}{y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right)}}{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y} \]
      associate-/l* [=>] 15.3	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\frac{y}{\frac{i + \left(c + \left(y + a\right) \cdot {y}^{2}\right) \cdot y}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}}} \]
      *-commutative [=>] 15.3	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \color{blue}{y \cdot \left(c + \left(y + a\right) \cdot {y}^{2}\right)}}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      *-commutative [=>] 15.3	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + y \cdot \left(c + \color{blue}{{y}^{2} \cdot \left(y + a\right)}\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      +-commutative [=>] 15.3	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + y \cdot \left(c + {y}^{2} \cdot \color{blue}{\left(a + y\right)}\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      distribute-lft-in [=>] 15.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \color{blue}{\left(y \cdot c + y \cdot \left({y}^{2} \cdot \left(a + y\right)\right)\right)}}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      +-commutative [<=] 15.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + y \cdot \left({y}^{2} \cdot \color{blue}{\left(y + a\right)}\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      associate-*r* [=>] 15.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + \color{blue}{\left(y \cdot {y}^{2}\right) \cdot \left(y + a\right)}\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      unpow2 [=>] 15.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(y + a\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      cube-mult [<=] 15.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{i + \left(y \cdot c + \color{blue}{{y}^{3}} \cdot \left(y + a\right)\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]

   5. Taylor expanded in y around inf 44.0%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\color{blue}{\left(\frac{y}{x} + \frac{a}{x}\right) - \frac{z}{{x}^{2}}}} \]

   6. Simplified 44.0%

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\color{blue}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{x \cdot x}\right)}} \]
      Proof

      [Start] 44.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\left(\frac{y}{x} + \frac{a}{x}\right) - \frac{z}{{x}^{2}}} \]
      associate--l+ [=>] 44.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\color{blue}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{{x}^{2}}\right)}} \]
      unpow2 [=>] 44.0	\[ \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{\color{blue}{x \cdot x}}\right)} \]

   if -8.5e21 < y < 6.0000000000000001e59

   1. Initial program 94.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} + \left(\frac{a \cdot x - z}{\frac{y}{\frac{a}{y}}} - \frac{a}{\frac{y}{x}}\right)\right)\right)\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} + \frac{y}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{x \cdot x}\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+59}:\\ \;\;\;\;\frac{t + y \cdot \left(y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right) + 230661.510616\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \left(\frac{z}{y} + \left(\frac{a \cdot x - z}{\frac{y}{\frac{a}{y}}} - \frac{a}{\frac{y}{x}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.6%
Cost	2764

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{t}{t_1} + \frac{y}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{x \cdot x}\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{t + y \cdot \left(y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right) + 230661.510616\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(\left(\left(x + \frac{27464.7644705}{y \cdot y}\right) + \frac{a}{y \cdot y} \cdot \left(a \cdot x - z\right)\right) - \left(x \cdot \frac{a}{y} + \frac{x}{\frac{y \cdot y}{b}}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	81.4%
Cost	2508

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ t_2 := \left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{t}{t_1} + \frac{y}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{x \cdot x}\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{t + y \cdot \left(y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right) + 230661.510616\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	77.4%
Cost	2376

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ t_2 := \left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -0.0022:\\ \;\;\;\;\frac{t}{t_1} + \frac{y}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{x \cdot x}\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(y \cdot y\right) \cdot z\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	77.0%
Cost	1993

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+70} \lor \neg \left(y \leq 7.5 \cdot 10^{+52}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(y \cdot y\right) \cdot z\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	73.5%
Cost	1865

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+71} \lor \neg \left(y \leq 5 \cdot 10^{+29}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	72.7%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+70} \lor \neg \left(y \leq 3.7 \cdot 10^{+28}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	65.4%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+70} \lor \neg \left(y \leq 2.6 \cdot 10^{+28}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	58.0%
Cost	1224

\[\begin{array}{l} t_1 := \left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{if}\;y \leq -3.75 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(y \cdot z\right)\right)}{y \cdot c}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	58.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+46} \lor \neg \left(y \leq 5.4 \cdot 10^{+26}\right):\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \end{array} \]

Alternative 10
Accuracy	53.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	50.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	27.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023140 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))