Difference of squares

Percentage Accurate: 94.1% → 100.0%

Time: 4.6s

Alternatives: 4

Speedup: 1.0×

• 42.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ a \cdot a - b \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) - (b * b)
end function

Java

public static double code(double a, double b) {
	return (a * a) - (b * b);
}

Python

def code(a, b):
	return (a * a) - (b * b)

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) - (b * b);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ a \cdot a - b \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) - (b * b)
end function

Java

public static double code(double a, double b) {
	return (a * a) - (b * b);
}

Python

def code(a, b):
	return (a * a) - (b * b)

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) - (b * b);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a - b\right) \cdot \left(a + b\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (- a b) (+ a b)))

C

double code(double a, double b) {
	return (a - b) * (a + b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a - b) * (a + b)
end function

Java

public static double code(double a, double b) {
	return (a - b) * (a + b);
}

Python

def code(a, b):
	return (a - b) * (a + b)

Julia

function code(a, b)
	return Float64(Float64(a - b) * Float64(a + b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a - b) * (a + b);
end

Wolfram

code[a_, b_] := N[(N[(a - b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[a \cdot a - b \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. difference-of-squares N/A

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(a - b\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(a - b\right) \cdot \color{blue}{\left(a + b\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(a - b\right), \color{blue}{\left(a + b\right)}\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, b\right), \left(\color{blue}{a} + b\right)\right) \]

   5. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, b\right), \mathsf{+.f64}\left(a, \color{blue}{b}\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(a - b\right) \cdot \left(a + b\right)} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+85}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+85) (* a a) (* b (- a b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+85) {
		tmp = a * a;
	} else {
		tmp = b * (a - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+85) then
        tmp = a * a
    else
        tmp = b * (a - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+85) {
		tmp = a * a;
	} else {
		tmp = b * (a - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+85:
		tmp = a * a
	else:
		tmp = b * (a - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+85)
		tmp = Float64(a * a);
	else
		tmp = Float64(b * Float64(a - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+85)
		tmp = a * a;
	else
		tmp = b * (a - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+85], N[(a * a), $MachinePrecision], N[(b * N[(a - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e85

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 79.6%

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

   if 1e85 < (*.f64 b b)

   1. Initial program 83.2%

      \[a \cdot a - b \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. difference-of-squares N/A

         \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(a - b\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(a - b\right) \cdot \color{blue}{\left(a + b\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(a - b\right), \color{blue}{\left(a + b\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, b\right), \left(\color{blue}{a} + b\right)\right) \]

      5. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, b\right), \mathsf{+.f64}\left(a, \color{blue}{b}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(a - b\right) \cdot \left(a + b\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(a, b\right), \color{blue}{b}\right) \]
   6. Step-by-step derivation

   7. Simplified 85.7%

      \[\leadsto \left(a - b\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+85}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;0 - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 3.8e-49) (- 0.0 (* b b)) (* a a)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 3.8e-49) {
		tmp = 0.0 - (b * b);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 3.8d-49) then
        tmp = 0.0d0 - (b * b)
    else
        tmp = a * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 3.8e-49) {
		tmp = 0.0 - (b * b);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 3.8e-49:
		tmp = 0.0 - (b * b)
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 3.8e-49)
		tmp = Float64(0.0 - Float64(b * b));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 3.8e-49)
		tmp = 0.0 - (b * b);
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 3.8e-49], N[(0.0 - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 3.7999999999999997e-49

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({b}^{2}\right) \]

      2. neg-sub0 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({b}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(b \cdot \color{blue}{b}\right)\right) \]

      5. *-lowering-*.f64 86.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   5. Simplified 86.4%

      \[\leadsto \color{blue}{0 - b \cdot b} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(b \cdot b\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(b \cdot b\right)\right) \]

      3. *-lowering-*.f64 86.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(b, b\right)\right) \]

   7. Applied egg-rr 86.4%

      \[\leadsto \color{blue}{-b \cdot b} \]

   if 3.7999999999999997e-49 < (*.f64 a a)

   1. Initial program 88.1%

      \[a \cdot a - b \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 78.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;0 - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.3% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* a a))

C

double code(double a, double b) {
	return a * a;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * a
end function

Java

public static double code(double a, double b) {
	return a * a;
}

Python

def code(a, b):
	return a * a

Julia

function code(a, b)
	return Float64(a * a)
end

MATLAB

function tmp = code(a, b)
	tmp = a * a;
end

Wolfram

code[a_, b_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[a \cdot a - b \cdot b \]
2. Add Preprocessing

3. Taylor expanded in a around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 58.1%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

5. Simplified 58.1%

   \[\leadsto \color{blue}{a \cdot a} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a + b\right) \cdot \left(a - b\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (+ a b) (- a b)))

C

double code(double a, double b) {
	return (a + b) * (a - b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a + b) * (a - b)
end function

Java

public static double code(double a, double b) {
	return (a + b) * (a - b);
}

Python

def code(a, b):
	return (a + b) * (a - b)

Julia

function code(a, b)
	return Float64(Float64(a + b) * Float64(a - b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a + b) * (a - b);
end

Wolfram

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024161 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :alt
  (! :herbie-platform default (* (+ a b) (- a b)))

  (- (* a a) (* b b)))