Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Average Error: 0.1 → 0.0

Time: 17.0s

Precision: binary64

Cost: 13632

Math

\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

↓

\[\mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, c - a \cdot \frac{b}{4}\right)\right) \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (fma x y (fma t (/ z 16.0) (- c (* a (/ b 4.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma(x, y, fma(t, (z / 16.0), (c - (a * (b / 4.0)))));
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

↓

function code(x, y, z, t, a, b, c)
	return fma(x, y, fma(t, Float64(z / 16.0), Float64(c - Float64(a * Float64(b / 4.0)))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_] := N[(x * y + N[(t * N[(z / 16.0), $MachinePrecision] + N[(c - N[(a * N[(b / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, c - a \cdot \frac{b}{4}\right)\right)} \]
   Proof

   [Start] 0.1	\[ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   associate--l+ [=>] 0.1	\[ \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]
   associate-+l+ [=>] 0.1	\[ \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(x, y, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]
   associate-+l- [=>] 0.1	\[ \mathsf{fma}\left(x, y, \color{blue}{\frac{z \cdot t}{16} - \left(\frac{a \cdot b}{4} - c\right)}\right) \]
   associate-*l/ [<=] 0.1	\[ \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{16} \cdot t} - \left(\frac{a \cdot b}{4} - c\right)\right) \]
   *-commutative [=>] 0.1	\[ \mathsf{fma}\left(x, y, \color{blue}{t \cdot \frac{z}{16}} - \left(\frac{a \cdot b}{4} - c\right)\right) \]
   fma-neg [=>] 0.1	\[ \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(t, \frac{z}{16}, -\left(\frac{a \cdot b}{4} - c\right)\right)}\right) \]
   neg-sub0 [=>] 0.1	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{0 - \left(\frac{a \cdot b}{4} - c\right)}\right)\right) \]
   associate-+l- [<=] 0.1	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{\left(0 - \frac{a \cdot b}{4}\right) + c}\right)\right) \]
   neg-sub0 [<=] 0.1	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{\left(-\frac{a \cdot b}{4}\right)} + c\right)\right) \]
   +-commutative [=>] 0.1	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{c + \left(-\frac{a \cdot b}{4}\right)}\right)\right) \]
   unsub-neg [=>] 0.1	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{c - \frac{a \cdot b}{4}}\right)\right) \]
   associate-*r/ [<=] 0.0	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, c - \color{blue}{a \cdot \frac{b}{4}}\right)\right) \]

3. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(x, y, \mathsf{fma}\left(t, \frac{z}{16}, c - a \cdot \frac{b}{4}\right)\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	7360

\[\mathsf{fma}\left(t, \frac{z}{16}, c - a \cdot \frac{b}{4}\right) + x \cdot y \]

Alternative 2
Error	28.3
Cost	2029

\[\begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625\\ t_2 := c + t_1\\ t_3 := x \cdot y + a \cdot \left(b \cdot -0.25\right)\\ t_4 := x \cdot y + t_1\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{+93}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-13}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.85 \cdot 10^{-98}:\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-141}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{-190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-223}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{-209} \lor \neg \left(a \leq 6 \cdot 10^{-147}\right) \land a \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	20.5
Cost	2008

\[\begin{array}{l} t_1 := c + \left(t \cdot z\right) \cdot 0.0625\\ t_2 := c + b \cdot \left(a \cdot -0.25\right)\\ t_3 := c + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -1000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-259}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 10^{-204}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 6 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	31.4
Cost	1636

\[\begin{array}{l} t_1 := c + x \cdot y\\ t_2 := c + \left(t \cdot z\right) \cdot 0.0625\\ t_3 := \left(a \cdot b\right) \cdot -0.25\\ \mathbf{if}\;a \leq -5.8 \cdot 10^{+173}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-189}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-199}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.3 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	36.0
Cost	1508

\[\begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625\\ t_2 := \left(a \cdot b\right) \cdot -0.25\\ \mathbf{if}\;c \leq -6 \cdot 10^{+46}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.15 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -6.8 \cdot 10^{-217}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-271}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 2.15 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]

Alternative 6
Error	5.7
Cost	1352

\[\begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625\\ \mathbf{if}\;a \cdot b \leq -1000:\\ \;\;\;\;\left(c + x \cdot y\right) + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{elif}\;a \cdot b \leq 200000000000:\\ \;\;\;\;c + \left(x \cdot y + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c + t_1\right) + \left(a \cdot b\right) \cdot -0.25\\ \end{array} \]

Alternative 7
Error	35.5
Cost	1244

\[\begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625\\ \mathbf{if}\;c \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq -1.65 \cdot 10^{-106}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-216}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-272}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \leq 7.8:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]

Alternative 8
Error	8.2
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+101} \lor \neg \left(a \cdot b \leq 10^{+42}\right):\\ \;\;\;\;c + b \cdot \left(a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(t \cdot z\right) \cdot 0.0625\right)\\ \end{array} \]

Alternative 9
Error	5.4
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1000 \lor \neg \left(a \cdot b \leq 6 \cdot 10^{+31}\right):\\ \;\;\;\;\left(c + x \cdot y\right) + a \cdot \left(b \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + \left(t \cdot z\right) \cdot 0.0625\right)\\ \end{array} \]

Alternative 10
Error	0.1
Cost	1088

\[\left(c - \frac{b}{\frac{4}{a}}\right) + \left(x \cdot y + \left(t \cdot z\right) \cdot 0.0625\right) \]

Alternative 11
Error	0.1
Cost	1088

\[c + \left(\left(\frac{t \cdot z}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) \]

Alternative 12
Error	23.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.4 \cdot 10^{+94} \lor \neg \left(a \cdot b \leq 2.5 \cdot 10^{+45}\right):\\ \;\;\;\;\left(a \cdot b\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 13
Error	36.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]

Alternative 14
Error	43.6
Cost	64

\[c \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))