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

Average Error: 16.5 → 0.0

Time: 3.7s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((1.0d0 - x) * (1.0d0 - y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (y * x)) - y
end function

Java

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

↓

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

Python

def code(x, y):
	return x + ((1.0 - x) * (1.0 - y))

↓

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

MATLAB

function tmp = code(x, y)
	tmp = x + ((1.0 - x) * (1.0 - y));
end

↓

function tmp = code(x, y)
	tmp = (1.0 + (y * x)) - 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[(1.0 + N[(y * x), $MachinePrecision]), $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
   (-.f64 (fma.f64 x y 1) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) 1)) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 y x)) 1) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 y))) x) 1) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (neg.f64 y))) x) 1) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 y) -1)) x) 1) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 y) (*.f64 -1 x))) 1) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 (neg.f64 y) (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) 1) y): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r-_binary64 (+.f64 (*.f64 (neg.f64 y) (neg.f64 x)) (-.f64 1 y))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> +-commutative_binary64 (+.f64 (-.f64 1 y) (*.f64 (neg.f64 y) (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 y))) (*.f64 (neg.f64 y) (neg.f64 x))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-+l+_binary64 (+.f64 1 (+.f64 (neg.f64 y) (*.f64 (neg.f64 y) (neg.f64 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 y) 1)) (*.f64 (neg.f64 y) (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite=> distribute-lft-out_binary64 (*.f64 (neg.f64 y) (+.f64 1 (neg.f64 x))))): 0 points increase in error, 2 points decrease in error
   (+.f64 1 (*.f64 (neg.f64 y) (Rewrite<= sub-neg_binary64 (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (neg.f64 x))) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 0 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x))) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 0 -1) x)) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite=> metadata-eval 0) x) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) x) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 x (*.f64 -1 x))) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 x (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 x (+.f64 (neg.f64 x) (+.f64 1 (*.f64 (neg.f64 y) (-.f64 1 x)))))): 74 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (neg.f64 x) 1) (*.f64 (neg.f64 y) (-.f64 1 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 x))) (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 x)) (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 1 x))) (*.f64 (neg.f64 y) (-.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite=> distribute-rgt-out_binary64 (*.f64 (-.f64 1 x) (+.f64 1 (neg.f64 y))))): 5 points increase in error, 0 points decrease in error
   (+.f64 x (*.f64 (-.f64 1 x) (Rewrite<= sub-neg_binary64 (-.f64 1 y)))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{\left(1 + y \cdot x\right) - y} \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	9.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0668657789567553 \cdot 10^{+21}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 13268111000.88214:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 2
Error	9.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0668657789567553 \cdot 10^{+21}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 236542032794271.16:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	19.3
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -0.12276131731217566:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 0.0032189468108935145:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 4
Error	18.4
Cost	192

\[1 - y \]

Alternative 5
Error	36.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(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))))