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

Percentage Accurate: 96.3% → 98.2%

Time: 12.3s

Precision: binary64

Cost: 19776

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

• 40.9% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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
 (fma c i (fma x y (fma z t (* a b)))))

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

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)
	return fma(c, i, fma(x, y, fma(z, t, Float64(a * b))))
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_] := N[(c * i + N[(x * y + N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

2. Simplified 98.8%

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

   [Start] 95.7%	\[ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   +-commutative [=>] 95.7%	\[ \color{blue}{c \cdot i + \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right)} \]
   fma-def [=>] 96.8%	\[ \color{blue}{\mathsf{fma}\left(c, i, \left(x \cdot y + z \cdot t\right) + a \cdot b\right)} \]
   associate-+l+ [=>] 96.8%	\[ \mathsf{fma}\left(c, i, \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)}\right) \]
   fma-def [=>] 98.0%	\[ \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(x, y, z \cdot t + a \cdot b\right)}\right) \]
   fma-def [=>] 98.8%	\[ \mathsf{fma}\left(c, i, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right)\right) \]

3. Final simplification 98.8%

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

Alternatives

Alternative 1
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 2
Accuracy	97.9%
Cost	13504

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

Alternative 3
Accuracy	98.0%
Cost	8260

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

Alternative 4
Accuracy	98.0%
Cost	7748

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

Alternative 5
Accuracy	97.6%
Cost	7492

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

Alternative 6
Accuracy	65.0%
Cost	2269

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ t_3 := a \cdot b + c \cdot i\\ \mathbf{if}\;c \cdot i \leq -5.4 \cdot 10^{+191}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \cdot i \leq -1.2 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq -2.3 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq -2.15 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 1.25 \cdot 10^{+59} \lor \neg \left(c \cdot i \leq 4.4 \cdot 10^{+112}\right) \land c \cdot i \leq 9.25 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Accuracy	64.9%
Cost	2269

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;c \cdot i \leq -1.6 \cdot 10^{+191}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -8.2 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq -5.4 \cdot 10^{-114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 5.5 \cdot 10^{+60} \lor \neg \left(c \cdot i \leq 1.4 \cdot 10^{+110}\right) \land c \cdot i \leq 9.25 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \end{array} \]

Alternative 8
Accuracy	64.7%
Cost	2268

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;c \cdot i \leq -4.9 \cdot 10^{+191}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq -4.6 \cdot 10^{-114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq -2.05 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 3.3 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 2.2 \cdot 10^{+114}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \end{array} \]

Alternative 9
Accuracy	97.6%
Cost	1988

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

Alternative 10
Accuracy	42.2%
Cost	1752

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -4.2 \cdot 10^{+194}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -1.6 \cdot 10^{+60}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq -1.8 \cdot 10^{-83}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.8 \cdot 10^{-108}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 4.9 \cdot 10^{+155}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 11
Accuracy	63.0%
Cost	1748

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;c \cdot i \leq -2.6 \cdot 10^{+203}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -5.3 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq -5.4 \cdot 10^{-114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq -7.2 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 6.4 \cdot 10^{+165}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 12
Accuracy	88.4%
Cost	1746

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.65 \cdot 10^{+111} \lor \neg \left(c \cdot i \leq 9.5 \cdot 10^{+60}\right) \land \left(c \cdot i \leq 1.9 \cdot 10^{+116} \lor \neg \left(c \cdot i \leq 9.25 \cdot 10^{+153}\right)\right):\\ \;\;\;\;c \cdot i + \left(a \cdot b + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \end{array} \]

Alternative 13
Accuracy	42.7%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -6 \cdot 10^{+130}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{-72}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 1.5 \cdot 10^{+69}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 14
Accuracy	61.8%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -6 \cdot 10^{+193}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -9 \cdot 10^{+79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 7 \cdot 10^{+163}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 15
Accuracy	85.4%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.7 \cdot 10^{+191}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 5.1 \cdot 10^{+163}:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \end{array} \]

Alternative 16
Accuracy	86.0%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -1.85 \cdot 10^{+191}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{elif}\;c \cdot i \leq 3.2 \cdot 10^{+160}:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \end{array} \]

Alternative 17
Accuracy	42.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.65 \cdot 10^{+114}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.1 \cdot 10^{+58}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 18
Accuracy	27.7%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023272 
(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)))