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

Average Error: 16.8 → 0.0

Time: 34.1s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = ((y * x) - y) - (-1.0d0)
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 ((y * x) - y) - -1.0;
}

Python

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

↓

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

MATLAB

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

↓

function tmp = code(x, y)
	tmp = ((y * x) - y) - -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[(N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision] - -1.0), $MachinePrecision]

Target

Original	16.8
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 16.8

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

2. Taylor expanded in x around -inf 0.0

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

3. Simplified 0.0

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

Alternatives

Alternative 1
Error	20.1
Cost	984

\[\begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+88}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-70}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+158}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+180}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 2
Error	0.9
Cost	840

\[\begin{array}{l} t_0 := \left(-1 + x\right) \cdot y\\ \mathbf{if}\;1 - y \leq -20000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;1 - y \leq 1.00002:\\ \;\;\;\;y \cdot x - -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.9
Cost	584

\[\begin{array}{l} t_0 := \left(-1 + x\right) \cdot y\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	9.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+35}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	19.1
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00115:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6
Error	36.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023033 
(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))))