Examples.Basics.BasicTests:f3 from sbv-4.4

Average Accuracy: 100.0% → 100.0%

Time: 3.0s

Precision: binary64

Cost: 704

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double y) {
	return (y * (y + (2.0 * x))) + (x * 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 + (2.0d0 * x))) + (x * 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 + (2.0 * x))) + (x * x);
}

Python

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[x_, y_] := N[(N[(y * N[(y + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 100.0%

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

3. Simplified 100.0%

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

   [Start] 100.0	\[ 2 \cdot \left(y \cdot x\right) + \left({y}^{2} + {x}^{2}\right) \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(2 \cdot \left(y \cdot x\right) + {y}^{2}\right) + {x}^{2}} \]
   unpow2 [=>] 100.0	\[ \left(2 \cdot \left(y \cdot x\right) + \color{blue}{y \cdot y}\right) + {x}^{2} \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(2, y \cdot x, y \cdot y\right)} + {x}^{2} \]
   unpow2 [=>] 100.0	\[ \mathsf{fma}\left(2, y \cdot x, y \cdot y\right) + \color{blue}{x \cdot x} \]

4. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \mathsf{fma}\left(2, y \cdot x, y \cdot y\right) + x \cdot x \]
   fma-udef [=>] 100.0	\[ \color{blue}{\left(2 \cdot \left(y \cdot x\right) + y \cdot y\right)} + x \cdot x \]
   *-commutative [=>] 100.0	\[ \left(2 \cdot \color{blue}{\left(x \cdot y\right)} + y \cdot y\right) + x \cdot x \]
   associate-*r* [=>] 100.0	\[ \left(\color{blue}{\left(2 \cdot x\right) \cdot y} + y \cdot y\right) + x \cdot x \]
   distribute-rgt-out [=>] 100.0	\[ \color{blue}{y \cdot \left(2 \cdot x + y\right)} + x \cdot x \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	87.0%
Cost	845

\[\begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-148} \lor \neg \left(y \leq 3.9 \cdot 10^{-106}\right) \land y \leq 1.35 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2
Accuracy	87.1%
Cost	845

\[\begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-148} \lor \neg \left(y \leq 8 \cdot 10^{-104}\right) \land y \leq 5.2 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + 2 \cdot x\right)\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	704

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

Alternative 4
Accuracy	86.8%
Cost	589

\[\begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-148} \lor \neg \left(y \leq 6 \cdot 10^{-104}\right) \land y \leq 2.45 \cdot 10^{-85}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5
Accuracy	100.0%
Cost	448

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

Alternative 6
Accuracy	56.3%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023130 
(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)))