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

Average Accuracy: 75.5% → 100.0%

Time: 4.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	75.5%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 75.5%

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

2. Simplified 100.0%

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

   [Start] 75.5	\[ x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   sub-neg [=>] 75.5	\[ x + \left(1 - x\right) \cdot \color{blue}{\left(1 + \left(-y\right)\right)} \]
   distribute-rgt-in [=>] 75.5	\[ x + \color{blue}{\left(1 \cdot \left(1 - x\right) + \left(-y\right) \cdot \left(1 - x\right)\right)} \]
   *-lft-identity [=>] 75.5	\[ x + \left(\color{blue}{\left(1 - x\right)} + \left(-y\right) \cdot \left(1 - x\right)\right) \]
   associate-+r+ [=>] 88.0	\[ \color{blue}{\left(x + \left(1 - x\right)\right) + \left(-y\right) \cdot \left(1 - x\right)} \]
   +-commutative [=>] 88.0	\[ \color{blue}{\left(\left(1 - x\right) + x\right)} + \left(-y\right) \cdot \left(1 - x\right) \]
   sub-neg [=>] 88.0	\[ \left(\color{blue}{\left(1 + \left(-x\right)\right)} + x\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   associate-+l+ [=>] 100.0	\[ \color{blue}{\left(1 + \left(\left(-x\right) + x\right)\right)} + \left(-y\right) \cdot \left(1 - x\right) \]
   neg-mul-1 [=>] 100.0	\[ \left(1 + \left(\color{blue}{-1 \cdot x} + x\right)\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   distribute-lft1-in [=>] 100.0	\[ \left(1 + \color{blue}{\left(-1 + 1\right) \cdot x}\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   metadata-eval [=>] 100.0	\[ \left(1 + \color{blue}{0} \cdot x\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   metadata-eval [<=] 100.0	\[ \left(1 + \color{blue}{\left(0 \cdot -1\right)} \cdot x\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   associate-*r* [<=] 100.0	\[ \left(1 + \color{blue}{0 \cdot \left(-1 \cdot x\right)}\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   neg-mul-1 [<=] 100.0	\[ \left(1 + 0 \cdot \color{blue}{\left(-x\right)}\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   mul0-lft [=>] 100.0	\[ \left(1 + \color{blue}{0}\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   metadata-eval [=>] 100.0	\[ \color{blue}{1} + \left(-y\right) \cdot \left(1 - x\right) \]
   sub-neg [=>] 100.0	\[ 1 + \left(-y\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)} \]
   distribute-rgt-in [=>] 100.0	\[ 1 + \color{blue}{\left(1 \cdot \left(-y\right) + \left(-x\right) \cdot \left(-y\right)\right)} \]
   *-lft-identity [=>] 100.0	\[ 1 + \left(\color{blue}{\left(-y\right)} + \left(-x\right) \cdot \left(-y\right)\right) \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(1 + \left(-y\right)\right) + \left(-x\right) \cdot \left(-y\right)} \]
   sub-neg [<=] 100.0	\[ \color{blue}{\left(1 - y\right)} + \left(-x\right) \cdot \left(-y\right) \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(-x\right) \cdot \left(-y\right) + \left(1 - y\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	56.4%
Cost	1776

\[\begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -205:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.18 \cdot 10^{-39}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -7.9 \cdot 10^{-184}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-271}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-285}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-126}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 3.45 \cdot 10^{-74}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-36}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Accuracy	85.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+65}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+51}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 3
Accuracy	85.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+63}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+51}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	448

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

Alternative 5
Accuracy	69.5%
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+19}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Accuracy	42.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023151 
(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))))