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

Average Error: 0.0 → 0.0

Time: 2.1s

Precision: binary64

Cost: 6720

Math

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

↓

\[\mathsf{fma}\left(y, 2, x \cdot x\right) \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (fma y 2.0 (* x x)))

C

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

↓

double code(double x, double y) {
	return fma(y, 2.0, (x * x));
}

Julia

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

↓

function code(x, y)
	return fma(y, 2.0, Float64(x * x))
end

Wolfram

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

↓

code[x_, y_] := N[(y * 2.0 + N[(x * x), $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. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{2 \cdot y + {x}^{2}} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 2, x \cdot x\right)} \]
   Proof
   (fma.f64 y 2 (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y 2 (Rewrite<= unpow2_binary64 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y 2) (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 y)) (pow.f64 x 2)): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(y, 2, x \cdot x\right) \]

Alternatives

Alternative 1
Error	9.8
Cost	584

\[\begin{array}{l} t_0 := y + x \cdot x\\ \mathbf{if}\;x \leq -2076697641788539400:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.2505131163667035 \cdot 10^{-30}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2076697641788539400:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.2505131163667035 \cdot 10^{-30}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	21.2
Cost	192

\[y + y \]

Alternative 5
Error	55.3
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022291 
(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))