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

Average Error: 0.0 → 0.0

Time: 4.5s

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	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   distribute-lft-out [=>] 0.0	\[ \color{blue}{x \cdot \left(2 + x\right)} + y \cdot y \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(x, 2 + x, y \cdot y\right)} \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(x, \color{blue}{x + 2}, y \cdot y\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	22.9
Cost	1248

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-168}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-241}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-231}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-153}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-124}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	1.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2.9 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 3
Error	4.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4
Error	0.0
Cost	704

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

Alternative 5
Error	9.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-39}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 3.57 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6
Error	0.0
Cost	576

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

Alternative 7
Error	24.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -115:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2750000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Error	44.6
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023031 
(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)))