Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 5

Speedup: 1.0×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.9e-89) (* x (+ x (* y 2.0))) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-89) {
		tmp = x * (x + (y * 2.0));
	} 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 (y <= 4.9d-89) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-89) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.9e-89:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.9e-89)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.9e-89)
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.9e-89], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9e-89

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*r* 66.4%

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

      3. *-commutative 66.4%

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

      4. unpow2 66.4%

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

      5. distribute-lft-in 67.5%

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

      6. *-commutative 67.5%

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

      7. fma-define 67.5%

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

   5. Simplified 67.5%

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

      1. fma-undefine 67.5%

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

   7. Applied egg-rr 67.5%

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

   if 4.9e-89 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.1%

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

   4. Taylor expanded in x around 0 71.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-91}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.1e-91) (* x (+ x y)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-91) {
		tmp = x * (x + y);
	} 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 (y <= 1.1d-91) then
        tmp = x * (x + y)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-91) {
		tmp = x * (x + y);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e-91:
		tmp = x * (x + y)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-91)
		tmp = Float64(x * Float64(x + y));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-91)
		tmp = x * (x + y);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e-91], N[(x * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e-91

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.2%

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

   if 1.1e-91 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 71.4%

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

   4. Taylor expanded in x around 0 70.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-91}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.5e-90) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-90) {
		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 (y <= 5.5d-90) 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 (y <= 5.5e-90) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e-90:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-90)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-90)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e-90], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000003e-90

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.2%

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

   4. Taylor expanded in x around inf 67.4%

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

   if 5.5000000000000003e-90 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 71.4%

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

   4. Taylor expanded in x around 0 70.7%

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

Alternative 5: 56.8% accurate, 2.3× 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 100.0%

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

3. Taylor expanded in x around inf 57.6%

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

4. Taylor expanded in x around inf 57.6%

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

Developer Target 1: 93.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024146 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* x x) (+ (* y y) (* 2 (* y x)))))

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