Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C

Average Error: 0.1 → 0.1

Time: 6.7s

Precision: binary64

Cost: 576

Math

\[x \cdot \left(y + z\right) + z \cdot 5 \]

↓

\[y \cdot x + z \cdot \left(5 + x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

↓

(FPCore (x y z) :precision binary64 (+ (* y x) (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

↓

double code(double x, double y, double z) {
	return (y * x) + (z * (5.0 + x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) + (z * 5.0d0)
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * x) + (z * (5.0d0 + x))
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

↓

public static double code(double x, double y, double z) {
	return (y * x) + (z * (5.0 + x));
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

↓

def code(x, y, z):
	return (y * x) + (z * (5.0 + x))

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

↓

function code(x, y, z)
	return Float64(Float64(y * x) + Float64(z * Float64(5.0 + x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

↓

function tmp = code(x, y, z)
	tmp = (y * x) + (z * (5.0 + x));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(y * x), $MachinePrecision] + N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(x + 5\right) \cdot z + x \cdot y \]

Derivation

1. Initial program 0.1

   \[x \cdot \left(y + z\right) + z \cdot 5 \]

2. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{\left(y + z\right) \cdot x + 5 \cdot z} \]

3. Simplified 0.1

   \[\leadsto \color{blue}{y \cdot x + z \cdot \left(5 + x\right)} \]
   Proof

   [Start] 0.1	\[ \left(y + z\right) \cdot x + 5 \cdot z \]
   rational_best.json-simplify-2 [<=] 0.1	\[ \color{blue}{x \cdot \left(y + z\right)} + 5 \cdot z \]
   rational_best.json-simplify-47 [<=] 0.1	\[ \color{blue}{\left(x \cdot z + x \cdot y\right)} + 5 \cdot z \]
   rational_best.json-simplify-2 [<=] 0.1	\[ \left(\color{blue}{z \cdot x} + x \cdot y\right) + 5 \cdot z \]
   rational_best.json-simplify-2 [<=] 0.1	\[ \left(z \cdot x + \color{blue}{y \cdot x}\right) + 5 \cdot z \]
   rational_best.json-simplify-1 [<=] 0.1	\[ \color{blue}{5 \cdot z + \left(z \cdot x + y \cdot x\right)} \]
   rational_best.json-simplify-43 [=>] 0.1	\[ \color{blue}{y \cdot x + \left(z \cdot x + 5 \cdot z\right)} \]
   rational_best.json-simplify-2 [=>] 0.1	\[ y \cdot x + \left(z \cdot x + \color{blue}{z \cdot 5}\right) \]
   rational_best.json-simplify-47 [=>] 0.1	\[ y \cdot x + \color{blue}{z \cdot \left(5 + x\right)} \]

4. Final simplification 0.1

   \[\leadsto y \cdot x + z \cdot \left(5 + x\right) \]

Alternatives

Alternative 1
Error	1.0
Cost	712

\[\begin{array}{l} t_0 := \left(z + y\right) \cdot x\\ \mathbf{if}\;x \leq -58000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;y \cdot x + 5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -58000000000:\\ \;\;\;\;z \cdot x + y \cdot x\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;y \cdot x + 5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) \cdot x\\ \end{array} \]

Alternative 3
Error	15.7
Cost	584

\[\begin{array}{l} t_0 := z \cdot \left(5 + x\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.2
Cost	584

\[\begin{array}{l} t_0 := z \cdot \left(5 + x\right)\\ \mathbf{if}\;z \leq -1.16 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.1
Cost	576

\[x \cdot \left(y + z\right) + z \cdot 5 \]

Alternative 6
Error	25.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-104}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]

Alternative 7
Error	34.7
Cost	192

\[5 \cdot z \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :herbie-target
  (+ (* (+ x 5.0) z) (* x y))

  (+ (* x (+ y z)) (* z 5.0)))