Difference of squares

Percentage Accurate: 94.2% → 100.0%

Time: 9.6s

Alternatives: 3

Speedup: 1.2×

• 43.9% 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: 94.2% 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, 1.2× 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 93.4%

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

   1. difference-of-squares N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a - b \cdot b \leq -5 \cdot 10^{-300}:\\ \;\;\;\;b \cdot \left(0 - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (- (* a a) (* b b)) -5e-300) (* b (- 0.0 b)) (* a a)))

C

double code(double a, double b) {
	double tmp;
	if (((a * a) - (b * b)) <= -5e-300) {
		tmp = b * (0.0 - 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) - (b * b)) <= (-5d-300)) then
        tmp = b * (0.0d0 - 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) - (b * b)) <= -5e-300) {
		tmp = b * (0.0 - b);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((a * a) - (b * b)) <= -5e-300:
		tmp = b * (0.0 - b)
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(a * a) - Float64(b * b)) <= -5e-300)
		tmp = Float64(b * Float64(0.0 - b));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((a * a) - (b * b)) <= -5e-300)
		tmp = b * (0.0 - b);
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], -5e-300], N[(b * N[(0.0 - b), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 a a) (*.f64 b b)) < -4.99999999999999996e-300

   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

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

      2. neg-sub0 N/A

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

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

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

      4. +-rgt-identity N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 98.5

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

   5. Simplified 98.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 98.5

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

   7. Applied egg-rr 98.5%

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

   if -4.99999999999999996e-300 < (-.f64 (*.f64 a a) (*.f64 b b))

   1. Initial program 88.7%

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

   3. Taylor expanded in a around inf

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

      1. +-rgt-identity N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 95.1

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

   5. Simplified 95.1%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 95.1

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

   7. Applied egg-rr 95.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a - b \cdot b \leq -5 \cdot 10^{-300}:\\ \;\;\;\;b \cdot \left(0 - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.0% 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 93.4%

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

3. Taylor expanded in a around inf

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

   1. +-rgt-identity N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 56.5

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

5. Simplified 56.5%

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

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 56.5

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

7. Applied egg-rr 56.5%

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

Developer Target 1: 100.0% accurate, 1.2× 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 2024195 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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