Average Error: 0.0 → 0
Time: 1.4s
Precision: binary64
Cost: 6656
\[x \cdot \left(x - 1\right) \]
↓
\[\mathsf{fma}\left(x, x, -x\right) \]
(FPCore (x) :precision binary64 (* x (- x 1.0)))
↓
(FPCore (x) :precision binary64 (fma x x (- x)))
double code(double x) {
	return x * (x - 1.0);
}
↓
double code(double x) {
	return fma(x, x, -x);
}

function code(x)
	return Float64(x * Float64(x - 1.0))
end
↓
function code(x)
	return fma(x, x, Float64(-x))
end

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

x \cdot \left(x - 1\right)
↓
\mathsf{fma}\left(x, x, -x\right)

Error

Target

Original0.0
Target0.0
Herbie0
\[x \cdot x - x \]

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{x \cdot x - x} \]
    Proof
    (-.f64 (*.f64 x x) x): 0 points increase in error, 0 points decrease in error
    (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 x x) (neg.f64 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 x x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 x x) (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 x x) (*.f64 1 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 x 1))): 5 points increase in error, 1 points decrease in error

  3. Applied egg-rr0

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

  4. Final simplification0

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

Alternatives

Alternative 1
Error1.7
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -7.202409520586058:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 0.06185657187349748:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 2
Error0.0
Cost320
\[x \cdot x - x \]
Alternative 3
Error21.4
Cost128
\[-x \]

Error

Reproduce

herbie shell --seed 2022316 
(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)))