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

Average Error: 0.1 → 0.1

Time: 13.5s

Precision: binary64

Cost: 960

Math

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

↓

\[\left(z \cdot \left(5 + x\right) + y \cdot \left(x \cdot 2\right)\right) - y \cdot x \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (+ (* z (+ 5.0 x)) (* y (* x 2.0))) (* y 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 ((z * (5.0 + x)) + (y * (x * 2.0))) - (y * 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 = ((z * (5.0d0 + x)) + (y * (x * 2.0d0))) - (y * 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 ((z * (5.0 + x)) + (y * (x * 2.0))) - (y * x);
}

Python

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

↓

def code(x, y, z):
	return ((z * (5.0 + x)) + (y * (x * 2.0))) - (y * 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(Float64(z * Float64(5.0 + x)) + Float64(y * Float64(x * 2.0))) - Float64(y * x))
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = ((z * (5.0 + x)) + (y * (x * 2.0))) - (y * 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[(N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * x), $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. Applied egg-rr 29.9

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left(\left(x \cdot \left(y + z\right) + z \cdot 5\right) \cdot \left(x \cdot \left(y + z\right) + z \cdot 5\right)\right)} \]

3. Taylor expanded in y around 0 43.4

   \[\leadsto \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \color{blue}{\left({\left(z \cdot x + 5 \cdot z\right)}^{2} + 2 \cdot \left(y \cdot \left(\left(z \cdot x + 5 \cdot z\right) \cdot x\right)\right)\right)} \]

4. Simplified 42.8

   \[\leadsto \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \color{blue}{\left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \left(\left(z \cdot \left(5 + x\right)\right) \cdot \left(y \cdot x\right)\right)\right)} \]
   Proof

   [Start] 43.4	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot x + 5 \cdot z\right)}^{2} + 2 \cdot \left(y \cdot \left(\left(z \cdot x + 5 \cdot z\right) \cdot x\right)\right)\right) \]
   rational.json-simplify-47 [=>] 43.4	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\color{blue}{\left(z \cdot \left(5 + x\right)\right)}}^{2} + 2 \cdot \left(y \cdot \left(\left(z \cdot x + 5 \cdot z\right) \cdot x\right)\right)\right) \]
   rational.json-simplify-2 [=>] 43.4	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(z \cdot x + 5 \cdot z\right)\right)}\right)\right) \]
   rational.json-simplify-47 [=>] 43.4	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(z \cdot \left(5 + x\right)\right)}\right)\right)\right) \]
   rational.json-simplify-43 [=>] 44.2	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \left(y \cdot \color{blue}{\left(z \cdot \left(\left(5 + x\right) \cdot x\right)\right)}\right)\right) \]
   rational.json-simplify-43 [<=] 43.4	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(z \cdot \left(5 + x\right)\right)\right)}\right)\right) \]
   rational.json-simplify-47 [<=] 43.4	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(z \cdot x + 5 \cdot z\right)}\right)\right)\right) \]
   rational.json-simplify-43 [<=] 42.8	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \color{blue}{\left(\left(z \cdot x + 5 \cdot z\right) \cdot \left(y \cdot x\right)\right)}\right) \]
   rational.json-simplify-47 [=>] 42.8	\[ \frac{1}{x \cdot \left(y + z\right) + z \cdot 5} \cdot \left({\left(z \cdot \left(5 + x\right)\right)}^{2} + 2 \cdot \left(\color{blue}{\left(z \cdot \left(5 + x\right)\right)} \cdot \left(y \cdot x\right)\right)\right) \]

5. Taylor expanded in z around inf 0.1

   \[\leadsto \color{blue}{\left(2 \cdot \left(y \cdot x\right) + z \cdot \left(5 + x\right)\right) - y \cdot x} \]

6. Simplified 0.1

   \[\leadsto \color{blue}{\left(z \cdot \left(5 + x\right) + y \cdot \left(x \cdot 2\right)\right) - y \cdot x} \]
   Proof

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

7. Final simplification 0.1

   \[\leadsto \left(z \cdot \left(5 + x\right) + y \cdot \left(x \cdot 2\right)\right) - y \cdot x \]

Alternatives

Alternative 1
Error	26.1
Cost	1248

\[\begin{array}{l} \mathbf{if}\;z \leq -0.72:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-65}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-79}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 10^{-122}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-100}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-73}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]

Alternative 2
Error	14.2
Cost	1112

\[\begin{array}{l} t_0 := z \cdot \left(5 + x\right)\\ t_1 := \left(z + y\right) \cdot x\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	25.4
Cost	984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-78}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-156}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-130}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-101}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-73}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]

Alternative 4
Error	16.3
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(5 + x\right)\\ \mathbf{if}\;z \leq -9.8 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-156}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.1
Cost	576

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

Alternative 6
Error	34.6
Cost	192

\[5 \cdot z \]

Reproduce

herbie shell --seed 2023077 
(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)))