Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 29.3 → 9.8

Time: 49.0s

Precision: binary64

Cost: 46281

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+64} \lor \neg \left(y \leq 6.5 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\\ \end{array} \]

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 -9.5e+64) (not (<= y 6.5e+67)))
   (+ (/ z y) (- x (/ a (/ y x))))
   (/
    (fma (fma (fma (fma x y z) y 27464.7644705) y 230661.510616) y t)
    (fma (fma (fma (+ y a) y b) y c) y 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 <= -9.5e+64) || !(y <= 6.5e+67)) {
		tmp = (z / y) + (x - (a / (y / x)));
	} else {
		tmp = fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma((y + a), y, b), y, c), y, 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 <= -9.5e+64) || !(y <= 6.5e+67))
		tmp = Float64(Float64(z / y) + Float64(x - Float64(a / Float64(y / x))));
	else
		tmp = Float64(fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(Float64(y + a), y, b), y, c), y, 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, -9.5e+64], N[Not[LessEqual[y, 6.5e+67]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] + N[(x - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * y + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(N[(y + a), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.50000000000000028e64 or 6.4999999999999995e67 < y

   1. Initial program 63.2

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

   2. Taylor expanded in y around inf 18.8

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

   3. Simplified 15.4

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

      [Start] 18.8	\[ \left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y} \]
      associate--l+ [=>] 18.8	\[ \color{blue}{\frac{z}{y} + \left(x - \frac{a \cdot x}{y}\right)} \]
      associate-/l* [=>] 15.4	\[ \frac{z}{y} + \left(x - \color{blue}{\frac{a}{\frac{y}{x}}}\right) \]

   if -9.50000000000000028e64 < y < 6.4999999999999995e67

   1. Initial program 5.9

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

   2. Simplified 5.9

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

      [Start] 5.9	\[ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 5.9	\[ \frac{\color{blue}{\mathsf{fma}\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 5.9	\[ \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705, y, 230661.510616\right)}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 5.9	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot y + z, y, 27464.7644705\right)}, y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 5.9	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, y, z\right)}, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 5.9	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c, y, i\right)}} \]
      fma-def [=>] 5.9	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(y + a\right) \cdot y + b, y, c\right)}, y, i\right)} \]
      fma-def [=>] 5.9	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y + a, y, b\right)}, y, c\right), y, i\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+64} \lor \neg \left(y \leq 6.5 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.8
Cost	2377

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+66} \lor \neg \left(y \leq 3.9 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	13.1
Cost	2249

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+64} \lor \neg \left(y \leq 5.5 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + x \cdot \left(y \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	12.5
Cost	2121

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+35} \lor \neg \left(y \leq 3.3 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)} \cdot \left(t + y \cdot \left(230661.510616 + z \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	15.5
Cost	1609

\[\begin{array}{l} \mathbf{if}\;y \leq -6100 \lor \neg \left(y \leq 1.25 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	17.9
Cost	1481

\[\begin{array}{l} \mathbf{if}\;y \leq -340 \lor \neg \left(y \leq 1.85 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + y \cdot 230661.510616\right) \cdot \frac{1}{i + y \cdot \left(c + a \cdot \left(y \cdot y\right)\right)}\\ \end{array} \]

Alternative 6
Error	16.7
Cost	1353

\[\begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{-6} \lor \neg \left(y \leq 8.6 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + y \cdot 230661.510616\right) \cdot \frac{1}{i + y \cdot \left(c + y \cdot b\right)}\\ \end{array} \]

Alternative 7
Error	24.0
Cost	1225

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-7} \lor \neg \left(y \leq 1.76 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)\right)}{i}\\ \end{array} \]

Alternative 8
Error	24.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-7} \lor \neg \left(y \leq 3.4 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \end{array} \]

Alternative 9
Error	24.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{-6} \lor \neg \left(y \leq 3.3 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i}\\ \end{array} \]

Alternative 10
Error	26.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-7} \lor \neg \left(y \leq 5.3 \cdot 10^{+28}\right):\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \end{array} \]

Alternative 11
Error	31.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-6} \lor \neg \left(y \leq 1.05 \cdot 10^{+33}\right):\\ \;\;\;\;x - x \cdot \frac{a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i}\\ \end{array} \]

Alternative 12
Error	29.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-6} \lor \neg \left(y \leq 2.35 \cdot 10^{+52}\right):\\ \;\;\;\;x - x \cdot \frac{a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i}\\ \end{array} \]

Alternative 13
Error	45.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+17}:\\ \;\;\;\;\frac{230661.510616}{c}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	31.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	46.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(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)))