Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.4% → 100.0%

Time: 1.1s

Alternatives: 3

Speedup: 2.0×

• 56.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	((x * x) + ((x * (2)) * y)) + (y * y)
END code

Initial Program: 93.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	((x * x) + ((x * (2)) * y)) + (y * y)
END code

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(y + x) * (y + x)
END code

Derivation

1. Initial program 93.4%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

3. Applied rewrites 100.0%

   \[\leadsto \left(y + x\right) \cdot \left(y + x\right) \]
4. Add Preprocessing

Alternative 2: 99.0% accurate, 2.0× speedup

Math

\[\mathsf{fma}\left(x, x, y \cdot y\right) \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fma x x (* y y)))

C

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

Julia

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(x * x) + (y * y)
END code

Derivation

1. Initial program 93.4%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

3. Applied rewrites 96.6%

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

4. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x, x, \left(2 \cdot x\right) \cdot y\right) \]
5. Step-by-step derivation

6. Applied rewrites 54.1%

   \[\leadsto \mathsf{fma}\left(x, x, \left(2 \cdot x\right) \cdot y\right) \]
7. Step-by-step derivation

8. Applied rewrites 54.1%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, x, y \cdot y\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(x, x, y \cdot y\right) \]
12. Add Preprocessing

Alternative 3: 57.2% accurate, 4.6× speedup

Math

\[y \cdot y \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* y y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	y * y
END code

Derivation

1. Initial program 93.4%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto y \cdot y \]
5. Step-by-step derivation

6. Applied rewrites 57.2%

   \[\leadsto y \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64
  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))