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

Average Error: 0.0 → 0.0

Time: 21.0s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (+ (* (- x -2.0) x) (* 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 - -2.0) * x) + (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 - (-2.0d0)) * x) + (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 - -2.0) * x) + (y * y);
}

Python

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

↓

def code(x, y):
	return ((x - -2.0) * x) + (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 - -2.0) * x) + 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 - -2.0) * x) + (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 - -2.0), $MachinePrecision] * x), $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. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

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

Alternatives

Alternative 1
Error	22.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-18}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Error	18.0
Cost	448

\[2 \cdot x + y \cdot y \]

Alternative 3
Error	42.2
Cost	192

\[2 \cdot x \]

Reproduce

herbie shell --seed 2023033 
(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)))