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

Average Accuracy: 100.0% → 100.0%

Time: 1.2s

Precision: binary64

Cost: 6656

Math

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

↓

\[\mathsf{fma}\left(x, x, -x\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (fma x x (- x)))

C

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

↓

double code(double x) {
	return fma(x, x, -x);
}

Julia

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

↓

function code(x)
	return fma(x, x, Float64(-x))
end

Wolfram

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

↓

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

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

\[x \cdot x - x \]

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot x - x} \]
   Proof

   [Start] 100.0	\[ x \cdot \left(x - 1\right) \]
   sub-neg [=>] 100.0	\[ x \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]
   distribute-rgt-in [=>] 100.0	\[ \color{blue}{x \cdot x + \left(-1\right) \cdot x} \]
   metadata-eval [=>] 100.0	\[ x \cdot x + \color{blue}{-1} \cdot x \]
   mul-1-neg [=>] 100.0	\[ x \cdot x + \color{blue}{\left(-x\right)} \]
   unsub-neg [=>] 100.0	\[ \color{blue}{x \cdot x - x} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -x\right)} \]
   Proof

   [Start] 100.0	\[ x \cdot x - x \]
   fma-neg [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, x, -x\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, x, -x\right) \]

Alternatives

Alternative 1
Accuracy	97.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	320

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

Alternative 3
Accuracy	100.0%
Cost	320

\[x \cdot x - x \]

Alternative 4
Accuracy	65.7%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023147 
(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)))