Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.7% → 98.9%

Time: 6.6s

Alternatives: 4

Speedup: 1.9×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

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

Alternative 1: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x \cdot x + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. +-commutative 93.3%

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

   2. associate-*l* 93.3%

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

   3. distribute-lft-out 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in y around 0 99.0%

   \[\leadsto x \cdot \color{blue}{x} + y \cdot y \]
6. Add Preprocessing

Alternative 2: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-46}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.8e-46) (* x x) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-46) {
		tmp = x * x;
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.8d-46)) then
        tmp = x * x
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-46) {
		tmp = x * x;
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.8e-46:
		tmp = x * x
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.8e-46)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.8e-46)
		tmp = x * x;
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.8e-46], N[(x * x), $MachinePrecision], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.8000000000000005e-46

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-*l* 91.9%

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

      3. distribute-lft-out 95.2%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in y around inf 92.1%

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

   6. Taylor expanded in y around 0 74.4%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right) + {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 74.4%

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

      2. unpow2 74.4%

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

      3. *-commutative 74.4%

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

      4. associate-*r* 74.4%

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

      5. distribute-lft-in 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in x around inf 78.1%

      \[\leadsto x \cdot \color{blue}{x} \]

   if -7.8000000000000005e-46 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 61.0%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right) + {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative 61.0%

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

      2. unpow2 61.0%

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

      3. associate-*r* 61.0%

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

      4. distribute-rgt-in 63.1%

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

      5. +-commutative 63.1%

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

      6. fma-undefine 63.1%

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

   5. Simplified 63.1%

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

      1. fma-undefine 63.1%

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

   7. Applied egg-rr 63.1%

      \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-46}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-46}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -9.5e-46) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-46) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.5d-46)) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-46) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.5e-46:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-46)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-46)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e-46], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999993e-46

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-*l* 91.9%

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

      3. distribute-lft-out 95.2%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in y around inf 92.1%

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

   6. Taylor expanded in y around 0 74.4%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right) + {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative 74.4%

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

      2. unpow2 74.4%

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

      3. *-commutative 74.4%

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

      4. associate-*r* 74.4%

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

      5. distribute-lft-in 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in x around inf 78.1%

      \[\leadsto x \cdot \color{blue}{x} \]

   if -9.49999999999999993e-46 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 61.0%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right) + {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative 61.0%

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

      2. unpow2 61.0%

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

      3. associate-*r* 61.0%

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

      4. distribute-rgt-in 63.1%

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

      5. +-commutative 63.1%

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

      6. fma-undefine 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in x around 0 64.2%

      \[\leadsto y \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

   1. +-commutative 93.3%

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

   2. associate-*l* 93.3%

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

   3. distribute-lft-out 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in y around inf 95.4%

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

6. Taylor expanded in y around 0 53.8%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right) + {x}^{2}} \]
7. Step-by-step derivation

   1. +-commutative 53.8%

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

   2. unpow2 53.8%

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

   3. *-commutative 53.8%

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

   4. associate-*r* 53.8%

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

   5. distribute-lft-in 57.7%

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

8. Simplified 57.7%

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

9. Taylor expanded in x around inf 58.4%

   \[\leadsto x \cdot \color{blue}{x} \]
10. Add Preprocessing

Developer target: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :alt
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))