Linear.V4:$cdot from linear-1.19.1.3, C

Percentage Accurate: 96.0% → 97.5%

Time: 18.3s

Precision: binary64

Cost: 8260

• Unsound rule application detected in e-graph. Results may not be sound.

• 41.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]

↓

\[\begin{array}{l} \mathbf{if}\;c \cdot i + \left(a \cdot b + \left(z \cdot t + x \cdot y\right)\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, y, z \cdot t\right) + \left(a \cdot b + c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (+ (* c i) (+ (* a b) (+ (* z t) (* x y)))) INFINITY)
   (+ (fma x y (* z t)) (+ (* a b) (* c i)))
   (fma y x (* z t))))

C

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

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) + ((a * b) + ((z * t) + (x * y)))) <= ((double) INFINITY)) {
		tmp = fma(x, y, (z * t)) + ((a * b) + (c * i));
	} else {
		tmp = fma(y, x, (z * t));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(Float64(z * t) + Float64(x * y)))) <= Inf)
		tmp = Float64(fma(x, y, Float64(z * t)) + Float64(Float64(a * b) + Float64(c * i)));
	else
		tmp = fma(y, x, Float64(z * t));
	end
	return tmp
end

Wolfram

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

↓

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i)) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]

   2. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
      associate-+l+ [=>] 100.0%	\[ \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]
      fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t\right)} + \left(a \cdot b + c \cdot i\right) \]

   if +inf.0 < (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i))

   1. Initial program 0.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]

   2. Simplified 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 0.0%	\[ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
      associate-+l+ [=>] 0.0%	\[ \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]
      associate-+l+ [=>] 0.0%	\[ \color{blue}{x \cdot y + \left(z \cdot t + \left(a \cdot b + c \cdot i\right)\right)} \]
      fma-def [=>] 20.0%	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t + \left(a \cdot b + c \cdot i\right)\right)} \]
      fma-def [=>] 40.0%	\[ \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b + c \cdot i\right)}\right) \]
      fma-def [=>] 50.0%	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

   3. Taylor expanded in a around 0 40.0%

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

   4. Taylor expanded in c around 0 50.5%

      \[\leadsto \color{blue}{y \cdot x + t \cdot z} \]

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, t \cdot z\right)} \]
      Step-by-step derivation

      [Start] 50.5%	\[ y \cdot x + t \cdot z \]
      fma-def [=>] 60.5%	\[ \color{blue}{\mathsf{fma}\left(y, x, t \cdot z\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i + \left(a \cdot b + \left(z \cdot t + x \cdot y\right)\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, y, z \cdot t\right) + \left(a \cdot b + c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.5%
Cost	8260

\[\begin{array}{l} \mathbf{if}\;c \cdot i + \left(a \cdot b + \left(z \cdot t + x \cdot y\right)\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, y, z \cdot t\right) + \left(a \cdot b + c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	19776

\[\mathsf{fma}\left(c, i, \mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, a \cdot b\right)\right)\right) \]

Alternative 3
Accuracy	97.6%
Cost	7748

\[\begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + \left(z \cdot t + x \cdot y\right)\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]

Alternative 4
Accuracy	97.5%
Cost	7748

\[\begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + \left(z \cdot t + x \cdot y\right)\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \end{array} \]

Alternative 5
Accuracy	96.9%
Cost	7232

\[\mathsf{fma}\left(c, i, z \cdot t + \left(a \cdot b + x \cdot y\right)\right) \]

Alternative 6
Accuracy	42.3%
Cost	2532

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.46 \cdot 10^{+113}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -7.8 \cdot 10^{-65}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -2.75 \cdot 10^{-174}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq -7.9 \cdot 10^{-177}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -2.2 \cdot 10^{-264}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{-194}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{-144}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 6 \cdot 10^{-31}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 4.8 \cdot 10^{+184}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 7
Accuracy	97.2%
Cost	1988

\[\begin{array}{l} t_1 := z \cdot t + x \cdot y\\ t_2 := c \cdot i + \left(a \cdot b + t_1\right)\\ \mathbf{if}\;t_2 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	62.8%
Cost	1748

\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ t_2 := a \cdot b + x \cdot y\\ \mathbf{if}\;c \cdot i \leq -2.1 \cdot 10^{+147}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -3.05 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 3.35 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 6 \cdot 10^{+179}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 9
Accuracy	65.3%
Cost	1748

\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ t_2 := a \cdot b + x \cdot y\\ t_3 := a \cdot b + c \cdot i\\ \mathbf{if}\;c \cdot i \leq -1.75 \cdot 10^{+120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \cdot i \leq -1.5 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 2.7 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 1.7 \cdot 10^{+179}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Accuracy	42.3%
Cost	1492

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.35 \cdot 10^{+109}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -2.1 \cdot 10^{-68}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 8.5 \cdot 10^{-195}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 8.6 \cdot 10^{-144}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 6.5 \cdot 10^{+132}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 11
Accuracy	87.7%
Cost	1484

\[\begin{array}{l} t_1 := a \cdot b + \left(c \cdot i + x \cdot y\right)\\ \mathbf{if}\;c \cdot i \leq -4 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 3.6 \cdot 10^{+41}:\\ \;\;\;\;a \cdot b + \left(z \cdot t + x \cdot y\right)\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+258}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \end{array} \]

Alternative 12
Accuracy	41.0%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.4 \cdot 10^{+140} \lor \neg \left(c \cdot i \leq -1.4 \cdot 10^{+81}\right) \land \left(c \cdot i \leq -1.75 \cdot 10^{-40} \lor \neg \left(c \cdot i \leq 7.5 \cdot 10^{+132}\right)\right):\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 13
Accuracy	76.2%
Cost	1232

\[\begin{array}{l} t_1 := a \cdot b + \left(c \cdot i + x \cdot y\right)\\ t_2 := c \cdot i + z \cdot t\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+258}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+147}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Accuracy	87.6%
Cost	1224

\[\begin{array}{l} t_1 := z \cdot t + x \cdot y\\ \mathbf{if}\;c \cdot i \leq -5.5 \cdot 10^{+119}:\\ \;\;\;\;a \cdot b + \left(c \cdot i + x \cdot y\right)\\ \mathbf{elif}\;c \cdot i \leq 7.4 \cdot 10^{+130}:\\ \;\;\;\;a \cdot b + t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + t_1\\ \end{array} \]

Alternative 15
Accuracy	60.6%
Cost	1108

\[\begin{array}{l} t_1 := c \cdot i + x \cdot y\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+111}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;x \leq -7.7 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{+62}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-274}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-109}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	61.0%
Cost	976

\[\begin{array}{l} t_1 := a \cdot b + c \cdot i\\ t_2 := z \cdot t + x \cdot y\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-205}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Accuracy	59.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.15 \cdot 10^{+147}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 3 \cdot 10^{-29}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 9 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 18
Accuracy	60.9%
Cost	844

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-274}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;x \leq 10^{-64}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Accuracy	27.7%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023277 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))