Average Error: 0.0 → 0
Time: 1.4s
Precision: binary64
Cost: 6720
\[\left(x \cdot x + y\right) + y \]
↓
\[\mathsf{fma}\left(x, x, y + y\right) \]
(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)

Error

Target

Original0.0
Target0.0
Herbie0
\[\left(y + y\right) + x \cdot x \]

Derivation

  1. Initial program 0.0

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

  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-def_binary64 (+.f64 (*.f64 x x) (+.f64 y y))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x x) y) y)): 0 points increase in error, 0 points decrease in error

  3. Final simplification0

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

Alternatives

Alternative 1
Error9.8
Cost850
\[\begin{array}{l} \mathbf{if}\;x \leq -230000000 \lor \neg \left(x \leq 4.5 \cdot 10^{-9}\right) \land \left(x \leq 2.5 \cdot 10^{+40} \lor \neg \left(x \leq 1.02 \cdot 10^{+58}\right)\right):\\ \;\;\;\;y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]
Alternative 2
Error38.0
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-99}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-81}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 3
Error11.2
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+34}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+58}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
Alternative 4
Error0.0
Cost448
\[y + \left(y + x \cdot x\right) \]
Alternative 5
Error55.3
Cost64
\[y \]

Error

Reproduce

herbie shell --seed 2022343 
(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))