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

Average Error: 16.5 → 0.0

Time: 3.5s

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	16.5
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 16.5

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

2. Simplified 0.0

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

   [Start] 16.5	\[ x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   *-commutative [=>] 16.5	\[ x + \color{blue}{\left(1 - y\right) \cdot \left(1 - x\right)} \]
   sub-neg [=>] 16.5	\[ x + \left(1 - y\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)} \]
   distribute-rgt-in [=>] 16.5	\[ x + \color{blue}{\left(1 \cdot \left(1 - y\right) + \left(-x\right) \cdot \left(1 - y\right)\right)} \]
   associate-+r+ [=>] 16.5	\[ \color{blue}{\left(x + 1 \cdot \left(1 - y\right)\right) + \left(-x\right) \cdot \left(1 - y\right)} \]
   +-commutative [=>] 16.5	\[ \color{blue}{\left(-x\right) \cdot \left(1 - y\right) + \left(x + 1 \cdot \left(1 - y\right)\right)} \]
   *-lft-identity [=>] 16.5	\[ \left(-x\right) \cdot \left(1 - y\right) + \left(x + \color{blue}{\left(1 - y\right)}\right) \]
   associate-+r+ [=>] 8.7	\[ \color{blue}{\left(\left(-x\right) \cdot \left(1 - y\right) + x\right) + \left(1 - y\right)} \]
   associate-+r- [=>] 8.7	\[ \color{blue}{\left(\left(\left(-x\right) \cdot \left(1 - y\right) + x\right) + 1\right) - y} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	18.8
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+41}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+69}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 2
Error	11.7
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+148}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 195:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+107}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+187}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Error	11.8
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+149}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-19}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+107}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+186}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Error	0.0
Cost	448

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

Alternative 5
Error	19.1
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 0.000205:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Error	36.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(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))))