Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.8 → 10.2

Time: 43.2s

Precision: binary64

Cost: 28676

Math

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

↓

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ t_2 := y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right)\\ t_3 := \frac{t_2 + t}{t_1}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;x \cdot \frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{+247}:\\ \;\;\;\;\frac{t}{t_1} + \frac{t_2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

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_3 (/ (+ t_2 t) t_1)))
   (if (<= t_3 (- INFINITY))
     (* x (/ (pow y 4.0) (fma y (fma y (fma y (+ y a) b) c) i)))
     (if (<= t_3 4e+247)
       (+ (/ t t_1) (/ t_2 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);
	double t_3 = (t_2 + t) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = x * (pow(y, 4.0) / fma(y, fma(y, fma(y, (y + a), b), c), i));
	} else if (t_3 <= 4e+247) {
		tmp = (t / t_1) + (t_2 / 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(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616))
	t_3 = Float64(Float64(t_2 + t) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(x * Float64((y ^ 4.0) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i)));
	elseif (t_3 <= 4e+247)
		tmp = Float64(Float64(t / t_1) + Float64(t_2 / 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[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + t), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(x * N[(N[Power[y, 4.0], $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+247], N[(N[(t / t$95$1), $MachinePrecision] + N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(z / y), $MachinePrecision] + N[(x - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}} \]
      Proof
      (/.f64 (fma.f64 (fma.f64 (fma.f64 (fma.f64 x y z) y 54929528941/2000000) y 28832688827/125000) y t) (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (fma.f64 (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) z)) y 54929528941/2000000) y 28832688827/125000) y t) (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000)) y 28832688827/125000) y t) (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000)) y t) (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t)) (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i)): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (fma.f64 (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 y a) y) b)) y c) y i)): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c)) y i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))): 0 points increase in error, 1 points decrease in error

   3. Taylor expanded in x around inf 63.4

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

   4. Simplified 28.2

      \[\leadsto \color{blue}{\frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \cdot x} \]
      Proof
      (*.f64 (/.f64 (pow.f64 y 4) (fma.f64 y (fma.f64 y (fma.f64 y (+.f64 y a) b) c) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 y (Rewrite<= metadata-eval (*.f64 2 2))) (fma.f64 y (fma.f64 y (fma.f64 y (+.f64 y a) b) c) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 y 2) (pow.f64 y 2))) (fma.f64 y (fma.f64 y (fma.f64 y (+.f64 y a) b) c) i)) x): 7 points increase in error, 9 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (pow.f64 y 2) 2)) (fma.f64 y (fma.f64 y (fma.f64 y (+.f64 y a) b) c) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (fma.f64 y (fma.f64 y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (+.f64 y a)) b)) c) i)) x): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (fma.f64 y (fma.f64 y (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 y a) y)) b) c) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (fma.f64 y (fma.f64 y (Rewrite<= fma-udef_binary64 (fma.f64 (+.f64 y a) y b)) c) i)) x): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (fma.f64 y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (fma.f64 (+.f64 y a) y b)) c)) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (fma.f64 y (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (fma.f64 (+.f64 y a) y b) y)) c) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (fma.f64 y (Rewrite<= fma-udef_binary64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c)) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (fma.f64 (fma.f64 (+.f64 y a) y b) y c)) i))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y)) i)) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (Rewrite<= fma-udef_binary64 (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 (pow.f64 y 2) 2) (/.f64 (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i) x))): 11 points increase in error, 9 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (fma.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y i))): 20 points increase in error, 10 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (fma.f64 (fma.f64 (+.f64 y a) y b) y c) y) i))): 0 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (fma.f64 (fma.f64 (+.f64 y a) y b) y c))) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (fma.f64 (+.f64 y a) y b) y) c))) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (*.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (+.f64 y a) y) b)) y) c)) i)): 1 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 a y)) y) b) y) c)) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (+.f64 a y))) b) y) c)) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (*.f64 y (+.f64 a y)))) y) c)) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 y (+.f64 b (*.f64 y (+.f64 a y))))) c)) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (*.f64 y (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 y (+.f64 a y)) b))) c)) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (*.f64 y (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 a y) y)) b)) c)) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (+.f64 (*.f64 y (+.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 y a)) y) b)) c)) i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 (pow.f64 y 2) 2) x) (+.f64 (*.f64 y (Rewrite<= +-commutative_binary64 (+.f64 c (*.f64 y (+.f64 (*.f64 (+.f64 y a) y) b))))) i)): 0 points increase in error, 0 points decrease in error

   if -inf.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)) < 3.99999999999999981e247

   1. Initial program 3.9

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

   2. Taylor expanded in t around inf 3.9

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

   if 3.99999999999999981e247 < (/.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 62.1

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

   2. Taylor expanded in y around inf 21.3

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

   3. Simplified 18.3

      \[\leadsto \color{blue}{\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)} \]
      Proof
      (+.f64 (/.f64 z y) (-.f64 x (/.f64 a (/.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 z y) (-.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a x) y)))): 16 points increase in error, 15 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 z y) x) (/.f64 (*.f64 a x) y))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} \leq -\infty:\\ \;\;\;\;x \cdot \frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} \leq 4 \cdot 10^{+247}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} + \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	3400

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ t_2 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{t}{t_1} + \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	10.5
Cost	2376

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.0
Cost	2124

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ t_2 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{t + x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}{t_2}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+38}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	14.7
Cost	1996

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ t_2 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{t + x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}{t_2}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+37}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.4
Cost	1864

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -31000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.7
Cost	1608

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -31500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.6
Cost	1484

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -17200000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot c}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	19.0
Cost	1224

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -29000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	22.3
Cost	1096

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -15000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{t}{i + y \cdot c} + \frac{y \cdot 230661.510616}{i}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	25.4
Cost	968

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -14600000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+24}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	25.4
Cost	968

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -15600000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	30.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -24500000000000:\\ \;\;\;\;x \cdot \left(1 - \frac{a}{y}\right)\\ \mathbf{elif}\;y \leq 2.22 \cdot 10^{+21}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	32.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -22000000000000:\\ \;\;\;\;x \cdot \left(1 - \frac{a}{y}\right)\\ \mathbf{elif}\;y \leq 0.205:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	32.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -17000000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.115:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	47.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))