Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3

Average Error: 0.0 → 0.0

Time: 10.6s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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) + ((x - 1.0d0) * z)
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 = (x * (y + z)) - z
end function

Java

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

↓

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

Python

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

↓

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

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(x - 1.0) * z))
end

↓

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

MATLAB

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

↓

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

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ \left(y + z\right) \cdot x + -1 \cdot z \]
   rational.json-simplify-1 [=>] 0.0	\[ \color{blue}{-1 \cdot z + \left(y + z\right) \cdot x} \]
   rational.json-simplify-2 [=>] 0.0	\[ -1 \cdot z + \color{blue}{x \cdot \left(y + z\right)} \]
   rational.json-simplify-51 [<=] 0.0	\[ -1 \cdot z + \color{blue}{\left(x \cdot z + y \cdot x\right)} \]
   rational.json-simplify-2 [<=] 0.0	\[ -1 \cdot z + \left(\color{blue}{z \cdot x} + y \cdot x\right) \]
   rational.json-simplify-2 [<=] 0.0	\[ -1 \cdot z + \left(z \cdot x + \color{blue}{x \cdot y}\right) \]
   rational.json-simplify-5 [<=] 0.0	\[ -1 \cdot z + \color{blue}{\left(\left(z \cdot x + x \cdot y\right) - 0\right)} \]
   metadata-eval [<=] 0.0	\[ -1 \cdot z + \left(\left(z \cdot x + x \cdot y\right) - \color{blue}{\left(0 - 0\right)}\right) \]
   rational.json-simplify-45 [<=] 0.0	\[ -1 \cdot z + \color{blue}{\left(0 - \left(0 - \left(z \cdot x + x \cdot y\right)\right)\right)} \]
   rational.json-simplify-12 [<=] 0.0	\[ -1 \cdot z + \left(0 - \color{blue}{\left(-\left(z \cdot x + x \cdot y\right)\right)}\right) \]
   rational.json-simplify-9 [<=] 0.0	\[ -1 \cdot z + \left(0 - \color{blue}{\left(z \cdot x + x \cdot y\right) \cdot -1}\right) \]
   rational.json-simplify-2 [=>] 0.0	\[ -1 \cdot z + \left(0 - \left(\color{blue}{x \cdot z} + x \cdot y\right) \cdot -1\right) \]
   rational.json-simplify-2 [=>] 0.0	\[ -1 \cdot z + \left(0 - \left(x \cdot z + \color{blue}{y \cdot x}\right) \cdot -1\right) \]
   rational.json-simplify-51 [=>] 0.0	\[ -1 \cdot z + \left(0 - \color{blue}{\left(x \cdot \left(y + z\right)\right)} \cdot -1\right) \]
   rational.json-simplify-2 [<=] 0.0	\[ -1 \cdot z + \left(0 - \color{blue}{\left(\left(y + z\right) \cdot x\right)} \cdot -1\right) \]
   rational.json-simplify-2 [<=] 0.0	\[ -1 \cdot z + \left(0 - \color{blue}{-1 \cdot \left(\left(y + z\right) \cdot x\right)}\right) \]
   rational.json-simplify-43 [<=] 0.0	\[ -1 \cdot z + \left(0 - \color{blue}{x \cdot \left(-1 \cdot \left(y + z\right)\right)}\right) \]
   rational.json-simplify-51 [<=] 0.0	\[ -1 \cdot z + \left(0 - x \cdot \color{blue}{\left(-1 \cdot z + y \cdot -1\right)}\right) \]
   rational.json-simplify-2 [<=] 0.0	\[ -1 \cdot z + \left(0 - x \cdot \left(-1 \cdot z + \color{blue}{-1 \cdot y}\right)\right) \]
   rational.json-simplify-2 [<=] 0.0	\[ -1 \cdot z + \left(0 - \color{blue}{\left(-1 \cdot z + -1 \cdot y\right) \cdot x}\right) \]
   rational.json-simplify-12 [<=] 0.0	\[ -1 \cdot z + \color{blue}{\left(-\left(-1 \cdot z + -1 \cdot y\right) \cdot x\right)} \]
   rational.json-simplify-9 [<=] 0.0	\[ -1 \cdot z + \color{blue}{\left(\left(-1 \cdot z + -1 \cdot y\right) \cdot x\right) \cdot -1} \]
   rational.json-simplify-2 [<=] 0.0	\[ -1 \cdot z + \color{blue}{-1 \cdot \left(\left(-1 \cdot z + -1 \cdot y\right) \cdot x\right)} \]

4. Taylor expanded in z around 0 0.0

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

5. Simplified 0.0

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

   [Start] 0.0	\[ y \cdot x + \left(x - 1\right) \cdot z \]
   rational.json-simplify-2 [=>] 0.0	\[ y \cdot x + \color{blue}{z \cdot \left(x - 1\right)} \]
   rational.json-simplify-16 [=>] 0.0	\[ y \cdot x + z \cdot \color{blue}{\left(x + -1\right)} \]
   rational.json-simplify-1 [=>] 0.0	\[ y \cdot x + z \cdot \color{blue}{\left(-1 + x\right)} \]
   rational.json-simplify-51 [<=] 0.0	\[ y \cdot x + \color{blue}{\left(z \cdot x + -1 \cdot z\right)} \]
   rational.json-simplify-2 [=>] 0.0	\[ y \cdot x + \left(z \cdot x + \color{blue}{z \cdot -1}\right) \]
   rational.json-simplify-8 [<=] 0.0	\[ y \cdot x + \left(z \cdot x + \color{blue}{\left(-z\right)}\right) \]
   rational.json-simplify-12 [=>] 0.0	\[ y \cdot x + \left(z \cdot x + \color{blue}{\left(0 - z\right)}\right) \]
   rational.json-simplify-48 [<=] 0.0	\[ y \cdot x + \color{blue}{\left(\left(0 + z \cdot x\right) - z\right)} \]
   rational.json-simplify-1 [<=] 0.0	\[ y \cdot x + \left(\color{blue}{\left(z \cdot x + 0\right)} - z\right) \]
   rational.json-simplify-4 [=>] 0.0	\[ y \cdot x + \left(\color{blue}{z \cdot x} - z\right) \]
   rational.json-simplify-48 [<=] 0.0	\[ \color{blue}{\left(z \cdot x + y \cdot x\right) - z} \]
   rational.json-simplify-1 [=>] 0.0	\[ \color{blue}{\left(y \cdot x + z \cdot x\right)} - z \]
   rational.json-simplify-2 [=>] 0.0	\[ \left(\color{blue}{x \cdot y} + z \cdot x\right) - z \]
   rational.json-simplify-51 [=>] 0.0	\[ \color{blue}{x \cdot \left(z + y\right)} - z \]
   rational.json-simplify-1 [=>] 0.0	\[ x \cdot \color{blue}{\left(y + z\right)} - z \]

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	23.6
Cost	852

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 350000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+42}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	14.2
Cost	848

\[\begin{array}{l} t_0 := \left(y + z\right) \cdot x\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-27}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-21}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	14.2
Cost	848

\[\begin{array}{l} t_0 := \left(y + z\right) \cdot x\\ t_1 := \left(x - 1\right) \cdot z\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	25.8
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-28}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-165}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	0.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot x\\ \end{array} \]

Alternative 6
Error	0.8
Cost	584

\[\begin{array}{l} t_0 := \left(y + z\right) \cdot x\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;y \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	35.1
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))