Data.Random.Distribution.Normal:normalTail from random-fu-0.2.6.2

Average Error: 0.0 → 0

Time: 1.5s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

Wolfram

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0

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

Derivation

1. Initial program 0.0

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

2. Simplified 0

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

   [Start] 0.0	\[ \left(x \cdot x + y\right) + y \]
   associate-+l+ [=>] 0.0	\[ \color{blue}{x \cdot x + \left(y + y\right)} \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(x, x, y + y\right)} \]

3. Final simplification 0

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

Alternatives

Alternative 1
Error	11.6
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+32}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-14} \lor \neg \left(x \leq -7.8 \cdot 10^{-64}\right) \land x \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	41.2
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023031 
(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))