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

Average Error: 24.85% → 0.01%

Time: 4.4s

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	24.85%
Target	0.01%
Herbie	0.01%

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

Derivation

1. Initial program 24.85

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

2. Simplified 0.01

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

   [Start] 24.85	\[ x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   +-commutative [=>] 24.85	\[ \color{blue}{\left(1 - x\right) \cdot \left(1 - y\right) + x} \]
   sub-neg [=>] 24.85	\[ \left(1 - x\right) \cdot \color{blue}{\left(1 + \left(-y\right)\right)} + x \]
   distribute-rgt-in [=>] 24.84	\[ \color{blue}{\left(1 \cdot \left(1 - x\right) + \left(-y\right) \cdot \left(1 - x\right)\right)} + x \]
   +-commutative [=>] 24.84	\[ \color{blue}{\left(\left(-y\right) \cdot \left(1 - x\right) + 1 \cdot \left(1 - x\right)\right)} + x \]
   sub-neg [=>] 24.84	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}\right) + x \]
   distribute-lft-in [=>] 24.84	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + \color{blue}{\left(1 \cdot 1 + 1 \cdot \left(-x\right)\right)}\right) + x \]
   metadata-eval [=>] 24.84	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + \left(\color{blue}{1} + 1 \cdot \left(-x\right)\right)\right) + x \]
   associate-+r+ [=>] 24.84	\[ \color{blue}{\left(\left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 1 \cdot \left(-x\right)\right)} + x \]
   associate-+l+ [=>] 0.01	\[ \color{blue}{\left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(1 \cdot \left(-x\right) + x\right)} \]
   neg-mul-1 [=>] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(1 \cdot \color{blue}{\left(-1 \cdot x\right)} + x\right) \]
   associate-*r* [=>] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(\color{blue}{\left(1 \cdot -1\right) \cdot x} + x\right) \]
   metadata-eval [=>] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \left(\color{blue}{-1} \cdot x + x\right) \]
   distribute-lft1-in [=>] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{\left(-1 + 1\right) \cdot x} \]
   metadata-eval [=>] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{0} \cdot x \]
   metadata-eval [<=] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{\left(0 \cdot -1\right)} \cdot x \]
   associate-*r* [<=] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{0 \cdot \left(-1 \cdot x\right)} \]
   metadata-eval [<=] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 0 \cdot \left(\color{blue}{\left(1 \cdot -1\right)} \cdot x\right) \]
   associate-*r* [<=] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 0 \cdot \color{blue}{\left(1 \cdot \left(-1 \cdot x\right)\right)} \]
   neg-mul-1 [<=] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + 0 \cdot \left(1 \cdot \color{blue}{\left(-x\right)}\right) \]
   mul0-lft [=>] 0.01	\[ \left(\left(-y\right) \cdot \left(1 - x\right) + 1\right) + \color{blue}{0} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	15.53%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -360000000000 \lor \neg \left(y \leq 1.65 \cdot 10^{-43}\right):\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 2
Error	43.35%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-24}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	15.25%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+48}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+90}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	0.01%
Cost	448

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

Alternative 5
Error	57.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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))))