Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3

Average Error: 16.4 → 0.0

Time: 3.0s

Precision: binary64

Cost: 6720

Math

\[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

↓

\[\mathsf{fma}\left(y, x + -1, 1\right) \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* (- 1.0 x) (- 1.0 y))))

↓

(FPCore (x y) :precision binary64 (fma y (+ x -1.0) 1.0))

C

double code(double x, double y) {
	return x + ((1.0 - x) * (1.0 - y));
}

↓

double code(double x, double y) {
	return fma(y, (x + -1.0), 1.0);
}

Julia

function code(x, y)
	return Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
end

↓

function code(x, y)
	return fma(y, Float64(x + -1.0), 1.0)
end

Wolfram

code[x_, y_] := N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]

Target

Original	16.4
Target	0.0
Herbie	0.0

\[y \cdot x - \left(y - 1\right) \]

Derivation

1. Initial program 16.4

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + -1, 1\right)} \]
   Proof

   [Start] 16.4	\[ x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   +-commutative [=>] 16.4	\[ \color{blue}{\left(1 - x\right) \cdot \left(1 - y\right) + x} \]
   sub-neg [=>] 16.4	\[ \left(1 - x\right) \cdot \color{blue}{\left(1 + \left(-y\right)\right)} + x \]
   distribute-rgt-in [=>] 16.4	\[ \color{blue}{\left(1 \cdot \left(1 - x\right) + \left(-y\right) \cdot \left(1 - x\right)\right)} + x \]
   +-commutative [=>] 16.4	\[ \color{blue}{\left(\left(-y\right) \cdot \left(1 - x\right) + 1 \cdot \left(1 - x\right)\right)} + x \]
   sub-neg [=>] 16.4	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}\right) + x \]
   distribute-lft-in [=>] 16.4	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + \color{blue}{\left(1 \cdot 1 + 1 \cdot \left(-x\right)\right)}\right) + x \]
   metadata-eval [=>] 16.4	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + \left(\color{blue}{1} + 1 \cdot \left(-x\right)\right)\right) + x \]
   associate-+r+ [=>] 16.4	\[ \color{blue}{\left(\left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 1 \cdot \left(-x\right)\right)} + x \]
   associate-+l+ [=>] 0.0	\[ \color{blue}{\left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(1 \cdot \left(-x\right) + x\right)} \]
   neg-mul-1 [=>] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(1 \cdot \color{blue}{\left(-1 \cdot x\right)} + x\right) \]
   associate-*r* [=>] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(\color{blue}{\left(1 \cdot -1\right) \cdot x} + x\right) \]
   metadata-eval [=>] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(\color{blue}{-1} \cdot x + x\right) \]
   distribute-lft1-in [=>] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{\left(-1 + 1\right) \cdot x} \]
   metadata-eval [=>] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{0} \cdot x \]
   metadata-eval [<=] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{\left(0 \cdot -1\right)} \cdot x \]
   associate-*r* [<=] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{0 \cdot \left(-1 \cdot x\right)} \]
   metadata-eval [<=] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 0 \cdot \left(\color{blue}{\left(1 \cdot -1\right)} \cdot x\right) \]
   associate-*r* [<=] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 0 \cdot \color{blue}{\left(1 \cdot \left(-1 \cdot x\right)\right)} \]
   neg-mul-1 [<=] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 0 \cdot \left(1 \cdot \color{blue}{\left(-x\right)}\right) \]
   mul0-lft [=>] 0.0	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{0} \]

3. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(y, x + -1, 1\right) \]

Alternatives

Alternative 1
Error	20.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+141}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-104}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+27}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 2
Error	9.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.38 \lor \neg \left(x \leq 1.55 \cdot 10^{+22}\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 3
Error	9.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+43}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+22}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	0.0
Cost	448

\[\left(1 + y \cdot x\right) - y \]

Alternative 5
Error	18.9
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Error	36.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1.0))

  (+ x (* (- 1.0 x) (- 1.0 y))))