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

Percentage Accurate: 95.7% → 97.8%

Time: 13.0s

Precision: binary64

Cost: 20548

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

• 41.9% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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 (<= (+ (* a b) (+ (* z t) (* x y))) INFINITY)
   (fma x y (fma z t (fma a b (* c i))))
   (+ (* c i) (fma z t (* x y)))))

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 (((a * b) + ((z * t) + (x * y))) <= ((double) INFINITY)) {
		tmp = fma(x, y, fma(z, t, fma(a, b, (c * i))));
	} else {
		tmp = (c * i) + fma(z, t, (x * y));
	}
	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(a * b) + Float64(Float64(z * t) + Float64(x * y))) <= Inf)
		tmp = fma(x, y, fma(z, t, fma(a, b, Float64(c * i))));
	else
		tmp = Float64(Float64(c * i) + fma(z, t, Float64(x * y)));
	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[(a * b), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x * y + N[(z * t + N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * i), $MachinePrecision] + N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.5%

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

   2. Simplified 99.2%

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

      [Start] 97.5%	\[ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
      associate-+l+ [=>] 97.5%	\[ \color{blue}{\left(x \cdot y + z \cdot t\right) + \left(a \cdot b + c \cdot i\right)} \]
      associate-+l+ [=>] 97.5%	\[ \color{blue}{x \cdot y + \left(z \cdot t + \left(a \cdot b + c \cdot i\right)\right)} \]
      fma-def [=>] 98.3%	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t + \left(a \cdot b + c \cdot i\right)\right)} \]
      fma-def [=>] 98.8%	\[ \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b + c \cdot i\right)}\right) \]
      fma-def [=>] 99.2%	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \color{blue}{\mathsf{fma}\left(a, b, c \cdot i\right)}\right)\right) \]

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

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0 40.0%

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

   3. Applied egg-rr 80.0%

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

      [Start] 40.0%	\[ c \cdot i + \left(y \cdot x + t \cdot z\right) \]
      +-commutative [=>] 40.0%	\[ c \cdot i + \color{blue}{\left(t \cdot z + y \cdot x\right)} \]
      *-commutative [=>] 40.0%	\[ c \cdot i + \left(\color{blue}{z \cdot t} + y \cdot x\right) \]
      *-commutative [<=] 40.0%	\[ c \cdot i + \left(z \cdot t + \color{blue}{x \cdot y}\right) \]
      fma-udef [<=] 80.0%	\[ c \cdot i + \color{blue}{\mathsf{fma}\left(z, t, x \cdot y\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	97.8%
Cost	20548

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

Alternative 2
Accuracy	97.6%
Cost	8260

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

Alternative 3
Accuracy	97.6%
Cost	8260

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

Alternative 4
Accuracy	97.5%
Cost	7748

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

Alternative 5
Accuracy	97.6%
Cost	7748

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

Alternative 6
Accuracy	41.6%
Cost	2272

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.3 \cdot 10^{+122}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5.8 \cdot 10^{+58}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq -4.6 \cdot 10^{-44}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -2.9 \cdot 10^{-185}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 2.9 \cdot 10^{-150}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 3.4 \cdot 10^{+82}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 5.4 \cdot 10^{+211}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 7
Accuracy	97.2%
Cost	1988

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

Alternative 8
Accuracy	63.5%
Cost	1490

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.25 \cdot 10^{+118} \lor \neg \left(c \cdot i \leq 1.8 \cdot 10^{-30}\right) \land \left(c \cdot i \leq 7.4 \cdot 10^{+80} \lor \neg \left(c \cdot i \leq 1.25 \cdot 10^{+212}\right)\right):\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 9
Accuracy	64.1%
Cost	1488

\[\begin{array}{l} t_1 := a \cdot b + c \cdot i\\ \mathbf{if}\;c \cdot i \leq -1.1 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 6.2 \cdot 10^{-150}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 5.6 \cdot 10^{+79}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	84.3%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -3.2 \cdot 10^{+121} \lor \neg \left(c \cdot i \leq 3 \cdot 10^{+212}\right):\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(z \cdot t + x \cdot y\right)\\ \end{array} \]

Alternative 11
Accuracy	87.1%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.8 \cdot 10^{+122} \lor \neg \left(c \cdot i \leq 1.1 \cdot 10^{+41}\right):\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(z \cdot t + x \cdot y\right)\\ \end{array} \]

Alternative 12
Accuracy	88.3%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+127} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+56}\right):\\ \;\;\;\;c \cdot i + \left(z \cdot t + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + x \cdot y\right)\\ \end{array} \]

Alternative 13
Accuracy	85.4%
Cost	1224

\[\begin{array}{l} t_1 := z \cdot t + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -6.8 \cdot 10^{+71}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;c \cdot i + t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + t_1\\ \end{array} \]

Alternative 14
Accuracy	41.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.22 \cdot 10^{+57}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 9.2 \cdot 10^{-122}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 5.2 \cdot 10^{+211}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 15
Accuracy	65.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.05 \cdot 10^{+117} \lor \neg \left(c \cdot i \leq 4 \cdot 10^{+42}\right):\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 16
Accuracy	62.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -5.9 \cdot 10^{+127}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4.8 \cdot 10^{+212}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 17
Accuracy	42.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.1 \cdot 10^{+178}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.65 \cdot 10^{+71}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 18
Accuracy	27.4%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023271 
(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)))