Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, A

Average Error: 10.4 → 0.2

Time: 4.9s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

(FPCore (x y) :precision binary64 (* (* (* x 3.0) x) y))

↓

(FPCore (x y) :precision binary64 (* (* y x) (* x 3.0)))

C

double code(double x, double y) {
	return ((x * 3.0) * x) * y;
}

↓

double code(double x, double y) {
	return (y * x) * (x * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 3.0d0) * x) * y
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * x) * (x * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((x * 3.0) * x) * y;
}

↓

public static double code(double x, double y) {
	return (y * x) * (x * 3.0);
}

Python

def code(x, y):
	return ((x * 3.0) * x) * y

↓

def code(x, y):
	return (y * x) * (x * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 3.0) * x) * y)
end

↓

function code(x, y)
	return Float64(Float64(y * x) * Float64(x * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 3.0) * x) * y;
end

↓

function tmp = code(x, y)
	tmp = (y * x) * (x * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]

↓

code[x_, y_] := N[(N[(y * x), $MachinePrecision] * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]

Target

Original	10.4
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 10.4

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

2. Simplified 0.3

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(3 \cdot y\right)\right)} \]
   Proof
   (*.f64 x (*.f64 x (*.f64 3 y))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 3) y))): 13 points increase in error, 26 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (*.f64 x 3)) y)): 60 points increase in error, 21 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 x 3) x)) y): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x around 0 0.2

   \[\leadsto x \cdot \color{blue}{\left(3 \cdot \left(y \cdot x\right)\right)} \]

4. Taylor expanded in x around 0 10.4

   \[\leadsto \color{blue}{3 \cdot \left(y \cdot {x}^{2}\right)} \]

5. Simplified 0.2

   \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot \left(3 \cdot x\right)} \]
   Proof
   (*.f64 (*.f64 y x) (*.f64 3 x)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 x) (*.f64 y x))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*l*_binary64 (*.f64 3 (*.f64 x (*.f64 y x)))): 31 points increase in error, 32 points decrease in error
   (*.f64 3 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 y x) x))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 x x)))): 49 points increase in error, 8 points decrease in error
   (*.f64 3 (*.f64 y (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	10.4
Cost	448

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

Reproduce

herbie shell --seed 2022290 
(FPCore (x y)
  :name "Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* (* x 3.0) (* x y))

  (* (* (* x 3.0) x) y))