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

Average Error: 0.0 → 0.0

Time: 3.7s

Precision: binary64

Cost: 6720

Math

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

↓

\[\left(-x\right) + {x}^{2} \]

FPCore

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

↓

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

C

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

↓

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

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 + (x ** 2.0d0)
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[x_] := N[((-x) + N[Power[x, 2.0], $MachinePrecision]), $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. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	320

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

Alternative 2
Error	22.0
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023075 
(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)))