Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 4

Speedup: 1.0×

• 44.3% 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. Final simplification 100.0%

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

Alternative 2: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.4e-100) (* x (+ x y)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-100) {
		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 <= 4.4d-100) 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 <= 4.4e-100) {
		tmp = x * (x + y);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e-100:
		tmp = x * (x + y)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-100)
		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 <= 4.4e-100)
		tmp = x * (x + y);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e-100], N[(x * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999978e-100

   1. Initial program 100.0%

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

      1. flip-+ 93.3%

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

      2. clear-num 93.1%

         \[\leadsto \left(x + y\right) \cdot \color{blue}{\frac{1}{\frac{x - y}{x \cdot x - y \cdot y}}} \]

      3. un-div-inv 93.2%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{x - y}{x \cdot x - y \cdot y}}} \]

      4. clear-num 93.0%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{x - y}}}} \]

      5. flip-+ 99.7%

         \[\leadsto \frac{x + y}{\frac{1}{\color{blue}{x + y}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{x + y}{\frac{1}{x + y}}} \]

   4. Taylor expanded in x around inf 68.0%

      \[\leadsto \frac{x + y}{\color{blue}{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. div-inv 68.0%

         \[\leadsto \color{blue}{\left(x + y\right) \cdot \frac{1}{\frac{1}{x}}} \]

      2. remove-double-div 68.1%

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

   6. Applied egg-rr 68.1%

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

   if 4.39999999999999978e-100 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.0%

      \[\leadsto \color{blue}{{y}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3: 69.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-71}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.1e-71) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-71) {
		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 <= 1.1d-71) 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 <= 1.1e-71) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e-71:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-71)
		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 <= 1.1e-71)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e-71], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.09999999999999999e-71

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 68.4%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 68.4%

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

   4. Simplified 68.4%

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

   if 1.09999999999999999e-71 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.9%

      \[\leadsto \color{blue}{{y}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 85.9%

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

   4. Simplified 85.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-71}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4: 57.7% 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. Taylor expanded in x around inf 55.9%

   \[\leadsto \color{blue}{{x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 55.9%

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

4. Simplified 55.9%

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

5. Final simplification 55.9%

   \[\leadsto x \cdot x \]

Developer target: 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 2023297 
(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)))