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

Average Error: 25.79% → 0.01%

Time: 3.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

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 Float64(fma(x, y, 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[(N[(x * y + 1.0), $MachinePrecision] - y), $MachinePrecision]

Target

Original	25.79%
Target	0.01%
Herbie	0.01%

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

Derivation

1. Initial program 25.79

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

2. Simplified 0.01

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

   [Start] 25.79	\[ x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   *-commutative [=>] 25.79	\[ x + \color{blue}{\left(1 - y\right) \cdot \left(1 - x\right)} \]
   sub-neg [=>] 25.79	\[ x + \left(1 - y\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)} \]
   distribute-rgt-in [=>] 25.79	\[ x + \color{blue}{\left(1 \cdot \left(1 - y\right) + \left(-x\right) \cdot \left(1 - y\right)\right)} \]
   associate-+r+ [=>] 25.79	\[ \color{blue}{\left(x + 1 \cdot \left(1 - y\right)\right) + \left(-x\right) \cdot \left(1 - y\right)} \]
   +-commutative [=>] 25.79	\[ \color{blue}{\left(-x\right) \cdot \left(1 - y\right) + \left(x + 1 \cdot \left(1 - y\right)\right)} \]
   *-lft-identity [=>] 25.79	\[ \left(-x\right) \cdot \left(1 - y\right) + \left(x + \color{blue}{\left(1 - y\right)}\right) \]
   associate-+r+ [=>] 13.6	\[ \color{blue}{\left(\left(-x\right) \cdot \left(1 - y\right) + x\right) + \left(1 - y\right)} \]
   associate-+r- [=>] 13.6	\[ \color{blue}{\left(\left(\left(-x\right) \cdot \left(1 - y\right) + x\right) + 1\right) - y} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	35.61%
Cost	984

\[\begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{+23}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-118}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.92 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+195}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 2
Error	15.31%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+36}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 3
Error	15.02%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+38}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 10000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Error	0.01%
Cost	448

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

Alternative 5
Error	30.1%
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -24000:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Error	56.29%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023088 
(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))))