Statistics.Sample:$swelfordMean from math-functions-0.1.5.2

Average Error: 0.0 → 0.0

Time: 4.9s

Precision: binary64

Cost: 576

Math

\[x + \frac{y - x}{z} \]

↓

\[\left(\frac{y}{z} + x\right) - \frac{x}{z} \]

FPCore

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

↓

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

C

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

↓

double code(double x, double y, double z) {
	return ((y / z) + x) - (x / 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) / 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 = ((y / z) + x) - (x / z)
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

   \[x + \frac{y - x}{z} \]

2. Taylor expanded in y around 0 0.0

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

3. Final simplification 0.0

   \[\leadsto \left(\frac{y}{z} + x\right) - \frac{x}{z} \]

Alternatives

Alternative 1
Error	24.1
Cost	984

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -32:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-76}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-161}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	24.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+53}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{elif}\;z \leq -32:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	12.0
Cost	584

\[\begin{array}{l} t_0 := x + \frac{y}{z}\\ \mathbf{if}\;z \leq -5 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-161}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	8.4
Cost	584

\[\begin{array}{l} t_0 := x + \frac{y}{z}\\ \mathbf{if}\;y \leq -600:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-32}:\\ \;\;\;\;x - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.9
Cost	584

\[\begin{array}{l} t_0 := x + \frac{y}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y - x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

\[x + \frac{y - x}{z} \]

Alternative 7
Error	34.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))