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

Average Error: 0.0 → 0.0

Time: 3.1s

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	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

   \[\leadsto \color{blue}{x \cdot \left(2 + x\right) + y \cdot y} \]
   Proof
   (+.f64 (*.f64 x (+.f64 2 x)) (*.f64 y y)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x 2) (*.f64 x x))) (*.f64 y y)): 2 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	22.9
Cost	1908

\[\begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-110}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq -1.18 \cdot 10^{-169}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-178}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-260}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-240}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-212}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-203}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-158}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-139}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-94}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-12}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 106:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	9.7
Cost	1100

\[\begin{array}{l} t_0 := x \cdot \left(x + 2\right)\\ \mathbf{if}\;y \cdot y \leq 5.9 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot y \leq 4 \cdot 10^{-103}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \cdot y \leq 1.1 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3
Error	4.4
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(x + 2\right)\\ \mathbf{if}\;x \leq -490:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.135:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	4.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{elif}\;x \leq 3.9:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]

Alternative 5
Error	1.3
Cost	712

\[\begin{array}{l} t_0 := y \cdot y + x \cdot x\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.086:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.0
Cost	576

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

Alternative 7
Error	25.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Error	42.2
Cost	192

\[x \cdot 2 \]

Reproduce

herbie shell --seed 2022325 
(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)))