Linear.Quaternion:$c/ from linear-1.19.1.3, E

Average Error: 0.1 → 0.1

Time: 10.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
   rational_best-simplify-1 [=>] 0.1	\[ \color{blue}{y \cdot y + \left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ y \cdot y + \color{blue}{\left(y \cdot y + \left(x \cdot x + y \cdot y\right)\right)} \]
   rational_best-simplify-43 [=>] 0.1	\[ \color{blue}{\left(x \cdot x + y \cdot y\right) + \left(y \cdot y + y \cdot y\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \color{blue}{\left(y \cdot y + y \cdot y\right) + \left(x \cdot x + y \cdot y\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \left(y \cdot y + y \cdot y\right) + \color{blue}{\left(y \cdot y + x \cdot x\right)} \]
   rational_best-simplify-43 [=>] 0.1	\[ \color{blue}{x \cdot x + \left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)} \]
   rational_best-simplify-47 [=>] 0.1	\[ x \cdot x + \left(y \cdot y + \color{blue}{y \cdot \left(y + y\right)}\right) \]
   rational_best-simplify-47 [=>] 0.1	\[ x \cdot x + \color{blue}{y \cdot \left(\left(y + y\right) + y\right)} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	576

\[x \cdot x + 3 \cdot \left(y \cdot y\right) \]

Alternative 2
Error	28.1
Cost	320

\[3 \cdot \left(y \cdot y\right) \]

Alternative 3
Error	28.0
Cost	320

\[y \cdot \left(3 \cdot y\right) \]

Reproduce

herbie shell --seed 2023094 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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