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

Average Error: 0.02% → 0%

Time: 1.8s

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	0.02%
Target	0.01%
Herbie	0%

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

Derivation

1. Initial program 0.02

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

2. Simplified 0

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

   [Start] 0.02	\[ \left(x \cdot x + y\right) + y \]
   associate-+l+ [=>] 0.01	\[ \color{blue}{x \cdot x + \left(y + y\right)} \]
   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	17.11%
Cost	850

\[\begin{array}{l} \mathbf{if}\;x \leq -0.000112 \lor \neg \left(x \leq 9 \cdot 10^{-57}\right) \land \left(x \leq 130000 \lor \neg \left(x \leq 6 \cdot 10^{+65}\right)\right):\\ \;\;\;\;y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]

Alternative 2
Error	18.49%
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-5}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-55} \lor \neg \left(x \leq 180000\right) \land x \leq 6 \cdot 10^{+65}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Error	59.48%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-64}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-112}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4
Error	0.02%
Cost	448

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

Alternative 5
Error	86.42%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023090 
(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))