Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 19.8s

Precision: binary64

Cost: 576

Math

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

↓

\[x + \frac{z}{\frac{\frac{1}{y}}{z}} \]

FPCore

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

↓

(FPCore (x y z) :precision binary64 (+ x (/ z (/ (/ 1.0 y) z))))

C

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

↓

double code(double x, double y, double z) {
	return x + (z / ((1.0 / y) / 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 * z) * 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 + (z / ((1.0d0 / y) / z))
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 5.8

   \[\leadsto x + \color{blue}{\frac{y}{\frac{\frac{1}{z}}{z}}} \]

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

   \[\leadsto x + \color{blue}{\frac{z}{\frac{1}{z \cdot y}}} \]

5. Taylor expanded in z around 0 0.1

   \[\leadsto x + \frac{z}{\color{blue}{\frac{1}{y \cdot z}}} \]

6. Simplified 0.1

   \[\leadsto x + \frac{z}{\color{blue}{\frac{\frac{1}{y}}{z}}} \]
   Proof

   [Start] 0.1	\[ x + \frac{z}{\frac{1}{y \cdot z}} \]
   rational.json-simplify-46 [=>] 0.1	\[ x + \frac{z}{\color{blue}{\frac{\frac{1}{y}}{z}}} \]

7. Final simplification 0.1

   \[\leadsto x + \frac{z}{\frac{\frac{1}{y}}{z}} \]

Alternatives

Alternative 1
Error	6.0
Cost	448

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

Alternative 2
Error	0.1
Cost	448

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

Alternative 3
Error	21.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))