Average Error: 0.0 → 0
Time: 1.1s
Precision: binary64
\[x \cdot \left(x - 1\right) \]
↓
\[\mathsf{fma}\left(x, x, -x\right) + \mathsf{fma}\left(-x, 1, x\right) \]
x \cdot \left(x - 1\right)
↓
\mathsf{fma}\left(x, x, -x\right) + \mathsf{fma}\left(-x, 1, x\right)

(FPCore (x) :precision binary64 (* x (- x 1.0)))
↓
(FPCore (x) :precision binary64 (+ (fma x x (- x)) (fma (- x) 1.0 x)))
double code(double x) {
	return x * (x - 1.0);
}
↓
double code(double x) {
	return fma(x, x, -x) + fma(-x, 1.0, x);
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in x around 0 0.0

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

  3. Applied *-un-lft-identity_binary640.0

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

  4. Applied unpow2_binary640.0

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

  5. Applied prod-diff_binary640

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -x \cdot 1\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)} \]

  6. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, x, -x\right) + \mathsf{fma}\left(-x, 1, x\right) \]

Reproduce

herbie shell --seed 2022068 
(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)))