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

Average Accuracy: 100.0% → 100.0%

Time: 3.7s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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(y, y, Float64(x * Float64(x + 2.0)))
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[(y * y + N[(x * N[(x + 2.0), $MachinePrecision]), $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%

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

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

3. Applied egg-rr 100.0%

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

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

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	83.7%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.32 \cdot 10^{-147} \lor \neg \left(y \cdot y \leq 3.9 \cdot 10^{-76}\right) \land y \cdot y \leq 6.8 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Accuracy	65.4%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-80}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-125}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-157}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-74}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3
Accuracy	93.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-9} \lor \neg \left(x \leq 0.022\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 4
Accuracy	98.2%
Cost	713

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

Alternative 5
Accuracy	100.0%
Cost	576

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

Alternative 6
Accuracy	61.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 0.038:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Accuracy	29.4%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023133 
(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)))