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

Average Accuracy: 100.0% → 100.0%

Time: 2.5s

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	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   distribute-lft-out [=>] 100.0	\[ \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, 2 + x, y \cdot y\right)} \]
   +-commutative [=>] 100.0	\[ \mathsf{fma}\left(x, \color{blue}{x + 2}, y \cdot y\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	93.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7.6:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]

Alternative 2
Accuracy	84.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-46} \lor \neg \left(y \leq 1.95 \cdot 10^{-41}\right):\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	576

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

Alternative 4
Accuracy	62.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+20}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 11000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Accuracy	65.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-48}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-86}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6
Accuracy	30.5%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023129 
(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)))