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

Average Error: 0.0 → 0.0

Time: 4.1s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

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 = (y * y) + (x * (x + 2.0d0))
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 (y * y) + (x * (x + 2.0));
}

Python

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

↓

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

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (y * y) + (x * (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[(N[(y * y), $MachinePrecision] + 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, 1 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	22.3
Cost	984

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-34}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{-86}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-86}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-276}:\\ \;\;\;\;x + x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-244}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-72}:\\ \;\;\;\;x + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Error	10.0
Cost	848

\[\begin{array}{l} t_0 := y \cdot y + \left(x + x\right)\\ t_1 := x \cdot \left(x + 2\right)\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3
Error	10.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-32}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Error	25.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -25500000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+43}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Error	45.0
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022338 
(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)))