Examples.Basics.BasicTests:f3 from sbv-4.4

Average Error: 0.04% → 0.03%

Time: 2.7s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

function code(x, y)
	return Float64(Float64(x + y) * Float64(x + y))
end

↓

function code(x, y)
	return Float64(Float64(y * Float64(y + x)) + Float64(x * Float64(y + x)))
end

MATLAB

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

↓

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

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(y * N[(y + x), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.04%
Target	0.03%
Herbie	0.03%

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

Derivation

1. Initial program 0.04

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

2. Applied egg-rr 0.03

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

3. Final simplification 0.03

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

Alternatives

Alternative 1
Error	0.04%
Cost	448

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

Alternative 2
Error	32.08%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-131}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3
Error	43.24%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

  (* (+ x y) (+ x y)))