Statistics.Correlation.Kendall:numOfTiesBy from math-functions-0.1.5.2

Average Error: 0.0 → 0.0

Time: 4.7s

Precision: binary64

Cost: 6656

Math

\[x \cdot \left(x - 1\right) \]

↓

\[{x}^{2} - x \]

FPCore

(FPCore (x) :precision binary64 (* x (- x 1.0)))

↓

(FPCore (x) :precision binary64 (- (pow x 2.0) x))

C

double code(double x) {
	return x * (x - 1.0);
}

↓

double code(double x) {
	return pow(x, 2.0) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x - 1.0d0)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) - x
end function

Java

public static double code(double x) {
	return x * (x - 1.0);
}

↓

public static double code(double x) {
	return Math.pow(x, 2.0) - x;
}

Python

def code(x):
	return x * (x - 1.0)

↓

def code(x):
	return math.pow(x, 2.0) - x

Julia

function code(x)
	return Float64(x * Float64(x - 1.0))
end

↓

function code(x)
	return Float64((x ^ 2.0) - x)
end

MATLAB

function tmp = code(x)
	tmp = x * (x - 1.0);
end

↓

function tmp = code(x)
	tmp = (x ^ 2.0) - x;
end

Wolfram

code[x_] := N[(x * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] - x), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0.0

\[x \cdot x - x \]

Derivation

1. Initial program 0.0

   \[x \cdot \left(x - 1\right) \]

2. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{-1 \cdot x + {x}^{2}} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\left(-x\right) + {x}^{2}} \]
   Proof

   [Start] 0.0	\[ -1 \cdot x + {x}^{2} \]
   rational.json-simplify-2 [=>] 0.0	\[ \color{blue}{x \cdot -1} + {x}^{2} \]
   rational.json-simplify-9 [=>] 0.0	\[ \color{blue}{\left(-x\right)} + {x}^{2} \]

4. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{-1 \cdot x + {x}^{2}} \]

5. Simplified 0.0

   \[\leadsto \color{blue}{{x}^{2} - x} \]
   Proof

   [Start] 0.0	\[ -1 \cdot x + {x}^{2} \]
   rational.json-simplify-1 [=>] 0.0	\[ \color{blue}{{x}^{2} + -1 \cdot x} \]
   rational.json-simplify-5 [<=] 0.0	\[ \color{blue}{\left({x}^{2} - 0\right)} + -1 \cdot x \]
   metadata-eval [<=] 0.0	\[ \left({x}^{2} - \color{blue}{\left(-1 - -1\right)}\right) + -1 \cdot x \]
   rational.json-simplify-44 [<=] 0.4	\[ \color{blue}{\left(-1 - \left(-1 - {x}^{2}\right)\right)} + -1 \cdot x \]
   rational.json-simplify-2 [=>] 0.4	\[ \left(-1 - \left(-1 - {x}^{2}\right)\right) + \color{blue}{x \cdot -1} \]
   rational.json-simplify-8 [<=] 0.4	\[ \left(-1 - \left(-1 - {x}^{2}\right)\right) + \color{blue}{\left(-x\right)} \]
   rational.json-simplify-12 [=>] 0.4	\[ \left(-1 - \left(-1 - {x}^{2}\right)\right) + \color{blue}{\left(0 - x\right)} \]
   metadata-eval [<=] 0.4	\[ \left(-1 - \left(-1 - {x}^{2}\right)\right) + \left(\color{blue}{\left(1 - 1\right)} - x\right) \]
   rational.json-simplify-46 [<=] 38.9	\[ \left(-1 - \left(-1 - {x}^{2}\right)\right) + \color{blue}{\left(1 - \left(1 + x\right)\right)} \]
   rational.json-simplify-1 [<=] 38.9	\[ \left(-1 - \left(-1 - {x}^{2}\right)\right) + \left(1 - \color{blue}{\left(x + 1\right)}\right) \]
   rational.json-simplify-64 [<=] 38.9	\[ \color{blue}{\left(-\left(-1 - {x}^{2}\right)\right) - \left(x + 1\right)} \]
   rational.json-simplify-13 [<=] 38.9	\[ \color{blue}{\left(0 - \left(-1 - {x}^{2}\right)\right)} - \left(x + 1\right) \]
   rational.json-simplify-44 [=>] 38.9	\[ \color{blue}{\left({x}^{2} - \left(-1 - 0\right)\right)} - \left(x + 1\right) \]
   metadata-eval [=>] 38.9	\[ \left({x}^{2} - \color{blue}{-1}\right) - \left(x + 1\right) \]
   rational.json-simplify-45 [=>] 38.7	\[ \color{blue}{{x}^{2} - \left(-1 + \left(x + 1\right)\right)} \]
   rational.json-simplify-41 [=>] 0.0	\[ {x}^{2} - \color{blue}{\left(x + \left(1 + -1\right)\right)} \]
   metadata-eval [=>] 0.0	\[ {x}^{2} - \left(x + \color{blue}{0}\right) \]
   rational.json-simplify-4 [=>] 0.0	\[ {x}^{2} - \color{blue}{x} \]

6. Final simplification 0.0

   \[\leadsto {x}^{2} - x \]

Alternatives

Alternative 1
Error	0.0
Cost	320

\[x \cdot \left(x - 1\right) \]

Alternative 2
Error	21.6
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "Statistics.Correlation.Kendall:numOfTiesBy from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (* x x) x)

  (* x (- x 1.0)))