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

Average Error: 0.0 → 0.0

Time: 2.5s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	10.3
Cost	848

\[\begin{array}{l} t_0 := y + x \cdot x\\ \mathbf{if}\;x \leq -2738921950194309600:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.776871919647458 \cdot 10^{-30}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq -5.947281636010873 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.12472732075115 \cdot 10^{-20}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	11.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -2738921950194309600:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -3.776871919647458 \cdot 10^{-30}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq -1.3535348695044549 \cdot 10^{-52}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7.12472732075115 \cdot 10^{-20}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	21.0
Cost	192

\[y + y \]

Alternative 5
Error	55.3
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022228 
(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))