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

Average Accuracy: 96.0% → 97.9%

Time: 12.0s

Precision: binary64

Cost: 19776

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

• 41.4% 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 93.3%

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

2. Simplified 98.4%

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

   [Start] 93.3	\[ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   +-commutative [=>] 93.3	\[ \color{blue}{c \cdot i + \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right)} \]
   fma-def [=>] 96.1	\[ \color{blue}{\mathsf{fma}\left(c, i, \left(x \cdot y + z \cdot t\right) + a \cdot b\right)} \]
   associate-+l+ [=>] 96.1	\[ \mathsf{fma}\left(c, i, \color{blue}{x \cdot y + \left(z \cdot t + a \cdot b\right)}\right) \]
   fma-def [=>] 96.9	\[ \mathsf{fma}\left(c, i, \color{blue}{\mathsf{fma}\left(x, y, z \cdot t + a \cdot b\right)}\right) \]
   fma-def [=>] 98.4	\[ \mathsf{fma}\left(c, i, \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, t, a \cdot b\right)}\right)\right) \]

3. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	97.2%
Cost	7753

\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -\infty \lor \neg \left(c \cdot i \leq 5 \cdot 10^{+271}\right):\\ \;\;\;\;\mathsf{fma}\left(c, i, x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, z \cdot t\right) + \left(a \cdot b + c \cdot i\right)\\ \end{array} \]

Alternative 2
Accuracy	96.8%
Cost	7752

\[\begin{array}{l} t_1 := x \cdot y + z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+247}:\\ \;\;\;\;\mathsf{fma}\left(c, i, t_1\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \end{array} \]

Alternative 3
Accuracy	64.9%
Cost	2268

\[\begin{array}{l} t_1 := a \cdot b + c \cdot i\\ t_2 := x \cdot y + z \cdot t\\ t_3 := c \cdot i + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -3.2 \cdot 10^{+172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -7.2 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -3.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -1.9 \cdot 10^{-307}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 6.3 \cdot 10^{-308}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 2.45 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 1.55 \cdot 10^{+36}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 4
Accuracy	42.0%
Cost	2012

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.7 \cdot 10^{+130}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.5 \cdot 10^{-307}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.4 \cdot 10^{-306}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 3.4 \cdot 10^{-191}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 3.3 \cdot 10^{-105}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.35 \cdot 10^{+44}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5
Accuracy	66.1%
Cost	2008

\[\begin{array}{l} t_1 := c \cdot i + z \cdot t\\ t_2 := a \cdot b + x \cdot y\\ t_3 := c \cdot i + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -1.1 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -6 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 1.1 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 5.6 \cdot 10^{+50}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 1.12 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	97.3%
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 7
Accuracy	37.8%
Cost	1776

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+31}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-50}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-53}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-275}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-244}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;y \leq 10^{-157}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+54}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+118}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+152}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+155}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8
Accuracy	87.9%
Cost	1484

\[\begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + x \cdot y\right)\\ \mathbf{if}\;a \cdot b \leq -5.6 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 1.12 \cdot 10^{+126}:\\ \;\;\;\;c \cdot i + \left(x \cdot y + z \cdot t\right)\\ \mathbf{elif}\;a \cdot b \leq 1.2 \cdot 10^{+277}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 9
Accuracy	54.3%
Cost	1372

\[\begin{array}{l} t_1 := a \cdot b + c \cdot i\\ t_2 := a \cdot b + x \cdot y\\ \mathbf{if}\;z \leq -7 \cdot 10^{+168}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{+37}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 10
Accuracy	49.5%
Cost	1248

\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ \mathbf{if}\;i \leq -1.6 \cdot 10^{-37}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;i \leq 4.1 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{+22}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;i \leq 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 2.15 \cdot 10^{+206}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{+226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+256}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;i \leq 3 \cdot 10^{+261}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]

Alternative 11
Accuracy	77.1%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+167} \lor \neg \left(z \leq -3.2 \cdot 10^{+124}\right) \land \left(z \leq -2.4 \cdot 10^{+44} \lor \neg \left(z \leq 3 \cdot 10^{+39}\right)\right):\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + x \cdot y\right)\\ \end{array} \]

Alternative 12
Accuracy	65.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.08 \cdot 10^{+40} \lor \neg \left(a \cdot b \leq 1.05 \cdot 10^{+111}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \end{array} \]

Alternative 13
Accuracy	42.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -6.6 \cdot 10^{+39}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{+45}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 14
Accuracy	27.1%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023160 
(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)))