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

Average Error: 0.02% → 0.02%

Time: 3.9s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + (x * 2.0d0)) + (y * y)
end function

Java

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

↓

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

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

↓

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

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

↓

function tmp = code(x, y)
	tmp = ((x * x) + (x * 2.0)) + (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[(N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $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. Final simplification 0.02

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

Alternatives

Alternative 1
Error	34.41%
Cost	1494

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.95 \cdot 10^{-199}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;y \cdot y \leq 1.3 \cdot 10^{-135} \lor \neg \left(y \cdot y \leq 2.6 \cdot 10^{-50}\right) \land \left(y \cdot y \leq 6.2 \cdot 10^{-31} \lor \neg \left(y \cdot y \leq 6.8 \cdot 10^{-10}\right)\right):\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	15.07%
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.02 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;y \cdot y \leq 5.6 \cdot 10^{-31}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \cdot y \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3
Error	6.45%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-9} \lor \neg \left(x \leq 3850000000000\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 4
Error	1.92%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 0.09\right):\\ \;\;\;\;x \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \end{array} \]

Alternative 5
Error	6.46%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{elif}\;x \leq 5400000000000:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]

Alternative 6
Error	0.02%
Cost	576

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

Alternative 7
Error	38.42%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3700000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Error	70.82%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023089 
(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)))