Examples.Basics.BasicTests:f1 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 4

Speedup: 1.0×

• 43.4% 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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) \cdot \left(x - y\right) \leq -1 \cdot 10^{-285}:\\ \;\;\;\;0 - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (+ x y) (- x y)) -1e-285) (- 0.0 (* y y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) * (x - y)) <= -1e-285) {
		tmp = 0.0 - (y * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x + y) * (x - y)) <= (-1d-285)) then
        tmp = 0.0d0 - (y * y)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) * (x - y)) <= -1e-285) {
		tmp = 0.0 - (y * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) * (x - y)) <= -1e-285:
		tmp = 0.0 - (y * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) * Float64(x - y)) <= -1e-285)
		tmp = Float64(0.0 - Float64(y * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) * (x - y)) <= -1e-285)
		tmp = 0.0 - (y * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], -1e-285], N[(0.0 - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x y) (-.f64 x y)) < -1.00000000000000007e-285

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - {y}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - {y}^{2}} \]

      4. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left({y}^{2} - 0\right)} \]

      5. sub-neg N/A

         \[\leadsto 0 - \color{blue}{\left({y}^{2} + \left(\mathsf{neg}\left(0\right)\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto 0 - \left(\color{blue}{y \cdot y} + \left(\mathsf{neg}\left(0\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 0 - \left(y \cdot y + \color{blue}{0}\right) \]

      8. accelerator-lowering-fma.f64 98.1

         \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(y, y, 0\right)} \]

   5. Simplified 98.1%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(y, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto 0 - \color{blue}{y \cdot y} \]

      2. *-lowering-*.f64 98.1

         \[\leadsto 0 - \color{blue}{y \cdot y} \]

   7. Applied egg-rr 98.1%

      \[\leadsto 0 - \color{blue}{y \cdot y} \]

   if -1.00000000000000007e-285 < (*.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 99.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 99.1

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

   7. Applied egg-rr 99.1%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) \cdot \left(x - y\right) \leq -1 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (+ x y) (- x y)) -1e-285) (* y (- x y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) * (x - y)) <= -1e-285) {
		tmp = y * (x - y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x + y) * (x - y)) <= (-1d-285)) then
        tmp = y * (x - y)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) * (x - y)) <= -1e-285) {
		tmp = y * (x - y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) * (x - y)) <= -1e-285:
		tmp = y * (x - y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) * Float64(x - y)) <= -1e-285)
		tmp = Float64(y * Float64(x - y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) * (x - y)) <= -1e-285)
		tmp = y * (x - y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], -1e-285], N[(y * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x y) (-.f64 x y)) < -1.00000000000000007e-285

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 97.1%

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

   if -1.00000000000000007e-285 < (*.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 99.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 99.1

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

   7. Applied egg-rr 99.1%

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

Alternative 4: 54.5% accurate, 2.0× 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

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{x}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + 0 \]

   3. accelerator-lowering-fma.f64 54.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 54.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 54.4

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

7. Applied egg-rr 54.4%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f1 from sbv-4.4"
  :precision binary64
  (* (+ x y) (- x y)))