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

Average Accuracy: 74.9% → 100.0%

Time: 2.8s

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	74.9%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 74.9%

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

2. Simplified 100.0%

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

   [Start] 74.9	\[ x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   sub-neg [=>] 74.9	\[ x + \left(1 - x\right) \cdot \color{blue}{\left(1 + \left(-y\right)\right)} \]
   distribute-rgt-in [=>] 75.0	\[ x + \color{blue}{\left(1 \cdot \left(1 - x\right) + \left(-y\right) \cdot \left(1 - x\right)\right)} \]
   *-lft-identity [=>] 75.0	\[ x + \left(\color{blue}{\left(1 - x\right)} + \left(-y\right) \cdot \left(1 - x\right)\right) \]
   associate-+r+ [=>] 87.3	\[ \color{blue}{\left(x + \left(1 - x\right)\right) + \left(-y\right) \cdot \left(1 - x\right)} \]
   +-commutative [=>] 87.3	\[ \color{blue}{\left(\left(1 - x\right) + x\right)} + \left(-y\right) \cdot \left(1 - x\right) \]
   sub-neg [=>] 87.3	\[ \left(\color{blue}{\left(1 + \left(-x\right)\right)} + x\right) + \left(-y\right) \cdot \left(1 - x\right) \]
   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	71.1%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+22}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 10^{-35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 2
Accuracy	83.8%
Cost	457

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+73} \lor \neg \left(x \leq 5 \cdot 10^{+112}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	70.3%
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 5
Accuracy	43.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023147 
(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))))