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

Average Accuracy: 100.0% → 100.0%

Time: 15.3s

Precision: binary64

Cost: 19776

Math

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

↓

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

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

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

Derivation

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%

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

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

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	19776

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

Alternative 2
Accuracy	63.8%
Cost	3568

\[\begin{array}{l} t_1 := x \cdot y + z \cdot t\\ t_2 := c \cdot i + z \cdot t\\ t_3 := c \cdot i + x \cdot y\\ t_4 := a \cdot b + c \cdot i\\ \mathbf{if}\;a \cdot b \leq -8.2 \cdot 10^{+18}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \cdot b \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1.45 \cdot 10^{-171}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq -6 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -7.5 \cdot 10^{-281}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-316}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-324}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 6.5 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 4.6 \cdot 10^{-51}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 1.95 \cdot 10^{-19}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 4.8 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 3
Accuracy	33.9%
Cost	2833

\[\begin{array}{l} \mathbf{if}\;i \leq -1.15 \cdot 10^{-18}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;i \leq -2.6 \cdot 10^{-99}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;i \leq -6.7 \cdot 10^{-220}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;i \leq -1.05 \cdot 10^{-272}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;i \leq -9.6 \cdot 10^{-294}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;i \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{-294}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;i \leq 1.05 \cdot 10^{-282}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{-275}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;i \leq 1.3 \cdot 10^{-257}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;i \leq 1.42 \cdot 10^{-227}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-170}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;i \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;i \leq 9.7 \cdot 10^{-23}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+23}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;i \leq 1.45 \cdot 10^{+32}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+53}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;i \leq 5.7 \cdot 10^{+94} \lor \neg \left(i \leq 1.35 \cdot 10^{+121}\right) \land i \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 4
Accuracy	38.4%
Cost	2532

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -9.5 \cdot 10^{-81}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -4.4 \cdot 10^{-257}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 5.5 \cdot 10^{-225}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 1.55 \cdot 10^{-127}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 3.4 \cdot 10^{-19}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 7.2 \cdot 10^{+57}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 5.3 \cdot 10^{+81}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 5
Accuracy	58.5%
Cost	2165

\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ t_2 := c \cdot i + z \cdot t\\ t_3 := a \cdot b + x \cdot y\\ \mathbf{if}\;i \leq -0.65:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -2.6 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq -3.1 \cdot 10^{-217}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -7.2 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 5 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 2400000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{+32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 1.05 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+91} \lor \neg \left(i \leq 4.8 \cdot 10^{+120}\right) \land i \leq 1.95 \cdot 10^{+146}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	58.3%
Cost	2032

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ t_3 := c \cdot i + x \cdot y\\ t_4 := c \cdot i + z \cdot t\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-87}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-296}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-223}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-104}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 0.00042:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+136}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+256}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	65.4%
Cost	2008

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ t_3 := a \cdot b + c \cdot i\\ \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \cdot i \leq -1.2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 0.0225:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 1.22 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Accuracy	47.7%
Cost	1638

\[\begin{array}{l} t_1 := a \cdot b + c \cdot i\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-231}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq -9.4 \cdot 10^{-304}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-283}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-206}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-73} \lor \neg \left(b \leq 4.2 \cdot 10^{-36} \lor \neg \left(b \leq 3.7 \cdot 10^{+18}\right) \land b \leq 2.45 \cdot 10^{+67}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 9
Accuracy	49.9%
Cost	1504

\[\begin{array}{l} t_1 := a \cdot b + z \cdot t\\ t_2 := a \cdot b + c \cdot i\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{-73}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-309}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 0.000225:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+158}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+257}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	86.0%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{+127}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+136}:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \end{array} \]

Alternative 11
Accuracy	100.0%
Cost	960

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

Alternative 12
Accuracy	40.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -3.2 \cdot 10^{+132}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 13
Accuracy	26.1%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023137 
(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)))