Numeric.Log:$clog1p from log-domain-0.10.2.1, A

Average Error: 0.02% → 0.02%

Time: 4.1s

Precision: binary64

Cost: 6848

Math

\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(x * N[(x + 2.0), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Target

Original	0.02%
Target	0.02%
Herbie	0.02%

\[y \cdot y + \left(2 \cdot x + x \cdot x\right) \]

Derivation

1. Initial program 0.02

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]

2. Simplified 0.02

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

   [Start] 0.02	\[ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   distribute-lft-out [=>] 0.02	\[ \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]
   fma-def [=>] 0.02	\[ \color{blue}{\mathsf{fma}\left(x, 2 + x, y \cdot y\right)} \]
   +-commutative [=>] 0.02	\[ \mathsf{fma}\left(x, \color{blue}{x + 2}, y \cdot y\right) \]

3. Final simplification 0.02

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

Alternatives

Alternative 1
Error	35.58%
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-137}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-247}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-301}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-100}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	5.67%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-13} \lor \neg \left(x \leq 0.00029\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 3
Error	1.96%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 1.8 \cdot 10^{-8}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 4
Error	15.78%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-23}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5
Error	0.02%
Cost	576

\[y \cdot y + x \cdot \left(x + 2\right) \]

Alternative 6
Error	37.95%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1400000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5400000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Error	69.7%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :herbie-target
  (+ (* y y) (+ (* 2.0 x) (* x x)))

  (+ (+ (* x 2.0) (* x x)) (* y y)))