Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 29.1 → 10.0

Time: 45.2s

Precision: binary64

Cost: 22217

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+35} \lor \neg \left(y \leq 4.4 \cdot 10^{+48}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} + y \cdot \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \mathsf{fma}\left(y, x, z\right)\right)}{\mathsf{fma}\left(y, c + y \cdot \mathsf{fma}\left(y + a, y, b\right), i\right)}\\ \end{array} \]

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
 (if (or (<= y -6.4e+35) (not (<= y 4.4e+48)))
   (+ x (- (/ z y) (* x (/ a y))))
   (+
    (/ t (+ i (* y (+ c (* y (+ (* y (+ y a)) b))))))
    (*
     y
     (/
      (+ 230661.510616 (* y (+ 27464.7644705 (* y (fma y x z)))))
      (fma y (+ c (* y (fma (+ y a) y b))) i))))))

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 tmp;
	if ((y <= -6.4e+35) || !(y <= 4.4e+48)) {
		tmp = x + ((z / y) - (x * (a / y)));
	} else {
		tmp = (t / (i + (y * (c + (y * ((y * (y + a)) + b)))))) + (y * ((230661.510616 + (y * (27464.7644705 + (y * fma(y, x, z))))) / fma(y, (c + (y * fma((y + a), y, b))), i)));
	}
	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)
	tmp = 0.0
	if ((y <= -6.4e+35) || !(y <= 4.4e+48))
		tmp = Float64(x + Float64(Float64(z / y) - Float64(x * Float64(a / y))));
	else
		tmp = Float64(Float64(t / Float64(i + Float64(y * Float64(c + Float64(y * Float64(Float64(y * Float64(y + a)) + b)))))) + Float64(y * Float64(Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * fma(y, x, z))))) / fma(y, Float64(c + Float64(y * fma(Float64(y + a), y, b))), i))));
	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_] := If[Or[LessEqual[y, -6.4e+35], N[Not[LessEqual[y, 4.4e+48]], $MachinePrecision]], N[(x + N[(N[(z / y), $MachinePrecision] - N[(x * N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(i + N[(y * N[(c + N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(y * x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * N[(c + N[(y * N[(N[(y + a), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.39999999999999965e35 or 4.3999999999999999e48 < y

   1. Initial program 61.4

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

   2. Taylor expanded in x around 0 61.8

      \[\leadsto \frac{\color{blue}{\left(y \cdot \left(230661.510616 + y \cdot \left(y \cdot z + 27464.7644705\right)\right) + {y}^{4} \cdot x\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Taylor expanded in y around inf 21.6

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

   4. Simplified 19.0

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

      [Start] 21.6	\[ \left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y} \]
      +-commutative [=>] 21.6	\[ \color{blue}{\left(x + \frac{z}{y}\right)} - \frac{a \cdot x}{y} \]
      associate--l+ [=>] 21.6	\[ \color{blue}{x + \left(\frac{z}{y} - \frac{a \cdot x}{y}\right)} \]
      associate-/l* [=>] 19.0	\[ x + \left(\frac{z}{y} - \color{blue}{\frac{a}{\frac{y}{x}}}\right) \]
      associate-/r/ [=>] 19.0	\[ x + \left(\frac{z}{y} - \color{blue}{\frac{a}{y} \cdot x}\right) \]

   if -6.39999999999999965e35 < y < 4.3999999999999999e48

   1. Initial program 2.9

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

   2. Taylor expanded in t around inf 2.9

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

   3. Applied egg-rr 2.8

      \[\leadsto \frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \color{blue}{\frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \mathsf{fma}\left(y, x, z\right)\right)}{\mathsf{fma}\left(y, c + y \cdot \mathsf{fma}\left(y + a, y, b\right), i\right)} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+35} \lor \neg \left(y \leq 4.4 \cdot 10^{+48}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} + y \cdot \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \mathsf{fma}\left(y, x, z\right)\right)}{\mathsf{fma}\left(y, c + y \cdot \mathsf{fma}\left(y + a, y, b\right), i\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.2
Cost	8841

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+35} \lor \neg \left(y \leq 6.1 \cdot 10^{+41}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + \left(y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)\right) + x \cdot {y}^{4}\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	2377

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+35} \lor \neg \left(y \leq 9 \cdot 10^{+41}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 3
Error	12.3
Cost	2252

\[\begin{array}{l} t_1 := i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)\\ t_2 := x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + x \cdot \left(y \cdot y\right)\right)\right)}{t_1}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+42}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	12.5
Cost	2121

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+35} \lor \neg \left(y \leq 5.2 \cdot 10^{+42}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 5
Error	13.0
Cost	1993

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+19} \lor \neg \left(y \leq 4.9 \cdot 10^{+42}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 6
Error	15.2
Cost	1865

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+60} \lor \neg \left(y \leq 2.2 \cdot 10^{+42}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 7
Error	16.2
Cost	1737

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+35} \lor \neg \left(y \leq 7.2 \cdot 10^{+32}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot c}\\ \end{array} \]

Alternative 8
Error	15.7
Cost	1609

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+24} \lor \neg \left(y \leq 9.2 \cdot 10^{+41}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 9
Error	23.9
Cost	1356

\[\begin{array}{l} t_1 := x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + b \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{t}{i + y \cdot c} + z \cdot \frac{y \cdot y}{c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	20.3
Cost	1353

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+35} \lor \neg \left(y \leq 1.8 \cdot 10^{+42}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)}\\ \end{array} \]

Alternative 11
Error	23.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+21} \lor \neg \left(y \leq 8 \cdot 10^{+22}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + b \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 12
Error	25.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+22} \lor \neg \left(y \leq 5.2 \cdot 10^{+22}\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;230661.510616 \cdot \frac{y}{i} + \frac{t}{i}\\ \end{array} \]

Alternative 13
Error	25.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+32} \lor \neg \left(y \leq 6500000000000\right):\\ \;\;\;\;x + \left(\frac{z}{y} - x \cdot \frac{a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i}\\ \end{array} \]

Alternative 14
Error	29.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00037:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+27}:\\ \;\;\;\;230661.510616 \cdot \frac{y}{i} + \frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	29.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -0.015:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	32.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -320000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	46.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(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)))