Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.6 → 9.2

Time: 50.9s

Precision: binary64

Cost: 70984

Math

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

↓

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ t_2 := \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{t_1}\\ t_3 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{\frac{c}{\frac{t_3}{y}} + \left(\frac{i}{t_3} + \frac{y}{\frac{\frac{t_3}{\mathsf{fma}\left(y + a, y, b\right)}}{y}}\right)}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(y \cdot \left(y \cdot \mathsf{fma}\left(x, y, z\right)\right) + y \cdot 27464.7644705\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

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
         (/
          (+
           (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
           t)
          t_1))
        (t_3 (fma y (fma y (fma y x z) 27464.7644705) 230661.510616)))
   (if (<= t_2 -1e-303)
     t_2
     (if (<= t_2 0.0)
       (/
        y
        (+
         (/ c (/ t_3 y))
         (+ (/ i t_3) (/ y (/ (/ t_3 (fma (+ y a) y b)) y)))))
       (if (<= t_2 4e+285)
         (/
          (+
           t
           (*
            y
            (+ 230661.510616 (+ (* y (* y (fma x y z))) (* y 27464.7644705)))))
          t_1)
         (+ (/ z y) (- x (/ a (/ y x)))))))))

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 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1;
	double t_3 = fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616);
	double tmp;
	if (t_2 <= -1e-303) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = y / ((c / (t_3 / y)) + ((i / t_3) + (y / ((t_3 / fma((y + a), y, b)) / y))));
	} else if (t_2 <= 4e+285) {
		tmp = (t + (y * (230661.510616 + ((y * (y * fma(x, y, z))) + (y * 27464.7644705))))) / t_1;
	} else {
		tmp = (z / y) + (x - (a / (y / x)));
	}
	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(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1)
	t_3 = fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616)
	tmp = 0.0
	if (t_2 <= -1e-303)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(y / Float64(Float64(c / Float64(t_3 / y)) + Float64(Float64(i / t_3) + Float64(y / Float64(Float64(t_3 / fma(Float64(y + a), y, b)) / y)))));
	elseif (t_2 <= 4e+285)
		tmp = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(Float64(y * Float64(y * fma(x, y, z))) + Float64(y * 27464.7644705))))) / t_1);
	else
		tmp = Float64(Float64(z / y) + Float64(x - Float64(a / Float64(y / x))));
	end
	return 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[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-303], t$95$2, If[LessEqual[t$95$2, 0.0], N[(y / N[(N[(c / N[(t$95$3 / y), $MachinePrecision]), $MachinePrecision] + N[(N[(i / t$95$3), $MachinePrecision] + N[(y / N[(N[(t$95$3 / N[(N[(y + a), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+285], N[(N[(t + N[(y * N[(230661.510616 + N[(N[(y * N[(y * N[(x * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 27464.7644705), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(z / y), $MachinePrecision] + N[(x - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < -9.99999999999999931e-304

   1. Initial program 3.6

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

   if -9.99999999999999931e-304 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 0.0

   1. Initial program 28.6

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

   2. Taylor expanded in t around 0 29.6

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

   3. Simplified 29.6

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

      [Start] 29.6	\[ \frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} \]
      *-commutative [<=] 29.6	\[ \frac{\color{blue}{y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right)}}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} \]
      associate-/l* [=>] 29.6	\[ \color{blue}{\frac{y}{\frac{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}}} \]
      fma-def [=>] 29.6	\[ \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(\left(y + a\right) \cdot y + b\right), i\right)}}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      +-commutative [=>] 29.6	\[ \frac{y}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right) + c}, i\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      *-commutative [=>] 29.6	\[ \frac{y}{\frac{\mathsf{fma}\left(y, \color{blue}{\left(\left(y + a\right) \cdot y + b\right) \cdot y} + c, i\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      fma-udef [<=] 29.6	\[ \frac{y}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\left(y + a\right) \cdot y + b, y, c\right)}, i\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      fma-def [=>] 29.6	\[ \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y + a, y, b\right)}, y, c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}} \]
      +-commutative [=>] 29.6	\[ \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), i\right)}{\color{blue}{y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right) + 230661.510616}}} \]

   4. Taylor expanded in c around 0 29.6

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

   5. Simplified 14.7

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

      [Start] 29.6	\[ \frac{y}{\frac{i}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)} + \left(\frac{c \cdot y}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)} + \frac{{y}^{2} \cdot \left(\left(y + a\right) \cdot y + b\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}\right)} \]
      +-commutative [<=] 29.6	\[ \frac{y}{\frac{i}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)} + \color{blue}{\left(\frac{{y}^{2} \cdot \left(\left(y + a\right) \cdot y + b\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)} + \frac{c \cdot y}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}\right)}} \]
      associate-+r+ [=>] 29.6	\[ \frac{y}{\color{blue}{\left(\frac{i}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)} + \frac{{y}^{2} \cdot \left(\left(y + a\right) \cdot y + b\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}\right) + \frac{c \cdot y}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}}} \]
      +-commutative [=>] 29.6	\[ \frac{y}{\color{blue}{\frac{c \cdot y}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)} + \left(\frac{i}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)} + \frac{{y}^{2} \cdot \left(\left(y + a\right) \cdot y + b\right)}{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}\right)}} \]

   if 0.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 3.9999999999999999e285

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   if 3.9999999999999999e285 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 63.4

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

   2. Taylor expanded in y around inf 20.9

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

   3. Simplified 17.6

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

      [Start] 20.9	\[ \left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y} \]
      associate--l+ [=>] 20.9	\[ \color{blue}{\frac{z}{y} + \left(x - \frac{a \cdot x}{y}\right)} \]
      associate-/l* [=>] 17.6	\[ \frac{z}{y} + \left(x - \color{blue}{\frac{a}{\frac{y}{x}}}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} \leq 0:\\ \;\;\;\;\frac{y}{\frac{c}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right)}{y}} + \left(\frac{i}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right)} + \frac{y}{\frac{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y + a, y, b\right)}}{y}}\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(y \cdot \left(y \cdot \mathsf{fma}\left(x, y, z\right)\right) + y \cdot 27464.7644705\right)\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	10692

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{t_1} \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(y \cdot \left(y \cdot \mathsf{fma}\left(x, y, z\right)\right) + y \cdot 27464.7644705\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 2
Error	10.2
Cost	4292

\[\begin{array}{l} t_1 := \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 3
Error	12.8
Cost	2121

\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+47} \lor \neg \left(y \leq 1.75 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{i + y \cdot \left(c + \left(y \cdot \left(y \cdot \left(y + a\right)\right) + y \cdot b\right)\right)}\\ \end{array} \]

Alternative 4
Error	12.8
Cost	1993

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+48} \lor \neg \left(y \leq 1.2 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 5
Error	14.1
Cost	1864

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616}{c + y \cdot \left(y \cdot \left(y + a\right) + b\right)}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.9
Cost	1864

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	20.6
Cost	1740

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{elif}\;y \leq 490000000:\\ \;\;\;\;\frac{y}{\frac{i + y \cdot \left(c + y \cdot b\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{\frac{y}{x} + \left(\frac{a}{x} - \frac{z}{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.1
Cost	1740

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-105}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+48}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	15.6
Cost	1609

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+44} \lor \neg \left(y \leq 6 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 10
Error	20.3
Cost	1353

\[\begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+44} \lor \neg \left(y \leq 1.45 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 11
Error	24.8
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y \leq -365000000 \lor \neg \left(y \leq 2.8 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{i}\\ \end{array} \]

Alternative 12
Error	25.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-13} \lor \neg \left(y \leq 0.00056\right):\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{x}{\frac{y}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \end{array} \]

Alternative 13
Error	25.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-13} \lor \neg \left(y \leq 0.00056\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \end{array} \]

Alternative 14
Error	25.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -365000000 \lor \neg \left(y \leq 0.00056\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i}\\ \end{array} \]

Alternative 15
Error	29.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -160000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 14000:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	46.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{230661.510616}{c}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	32.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -43000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 115000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	47.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(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)))