\[\left(x \cdot x + y\right) + y \]

↓

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

(FPCore (x y) :precision binary64 (+ (+ (* x x) y) y))

↓

(FPCore (x y) :precision binary64 (fma x x (+ y y)))

double code(double x, double y) {
	return ((x * x) + y) + y;
}

↓

double code(double x, double y) {
	return fma(x, x, (y + y));
}

function code(x, y)
	return Float64(Float64(Float64(x * x) + y) + y)
end

↓

function code(x, y)
	return fma(x, x, Float64(y + y))
end

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

↓

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

\left(x \cdot x + y\right) + y

↓

\mathsf{fma}\left(x, x, y + y\right)

Original	0.0
Target	0.0
Herbie	0

Derivation

Initial program 0.0
\[\left(x \cdot x + y\right) + y \]
Taylor expanded in x around 0 0.0
\[\leadsto \color{blue}{2 \cdot y + {x}^{2}} \]
Simplified0
\[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y + y\right)} \]
Proof
(fma.f64 x x (+.f64 y y)): 0 points increase in error, 0 points decrease in error
(Rewrite=> fma-udef_binary64 (+.f64 (*.f64 x x) (+.f64 y y))): 2 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 (+.f64 y y) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite=> count-2_binary64 (*.f64 2 y)) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 2 y) (Rewrite<= unpow2_binary64 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
Final simplification0
\[\leadsto \mathsf{fma}\left(x, x, y + y\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	6720

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

Alternative 2
Error	11.6
Cost	1492

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.5 \cdot 10^{-55}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \cdot x \leq 160:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 1.42 \cdot 10^{+38}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \cdot x \leq 7 \cdot 10^{+60}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 9 \cdot 10^{+94}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Error	10.6
Cost	584

\[\begin{array}{l} t_0 := y + x \cdot x\\ \mathbf{if}\;x \leq -1.76 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-31}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	39.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-26}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{-54}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Error	0.0
Cost	448

\[y + \left(y + x \cdot x\right) \]

Alternative 6
Error	55.3
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022330 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2"
  :precision binary64

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

  (+ (+ (* x x) y) y))

Error

Target

Derivation

Alternatives

Error

Reproduce