Difference of squares

Percentage Accurate: 94.5% → 100.0%

Time: 3.0s

Alternatives: 3

Speedup: 1.2×

• 42.8% 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: 94.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot a - b \cdot b\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-306} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-b\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* a a) (* b b))))
   (if (or (<= t_0 -4e-306) (not (<= t_0 INFINITY))) (* (- b) b) (* a a))))

C

double code(double a, double b) {
	double t_0 = (a * a) - (b * b);
	double tmp;
	if ((t_0 <= -4e-306) || !(t_0 <= ((double) INFINITY))) {
		tmp = -b * b;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (a * a) - (b * b);
	double tmp;
	if ((t_0 <= -4e-306) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -b * b;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) - (b * b)
	tmp = 0
	if (t_0 <= -4e-306) or not (t_0 <= math.inf):
		tmp = -b * b
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) - Float64(b * b))
	tmp = 0.0
	if ((t_0 <= -4e-306) || !(t_0 <= Inf))
		tmp = Float64(Float64(-b) * b);
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) - (b * b);
	tmp = 0.0;
	if ((t_0 <= -4e-306) || ~((t_0 <= Inf)))
		tmp = -b * b;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-306], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-b) * b), $MachinePrecision], N[(a * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 a a) (*.f64 b b)) < -4.00000000000000011e-306 or +inf.0 < (-.f64 (*.f64 a a) (*.f64 b b))

   1. Initial program 87.6%

      \[a \cdot a - b \cdot b \]
   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. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -4.00000000000000011e-306 < (-.f64 (*.f64 a a) (*.f64 b b)) < +inf.0

   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. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 12.7

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a - b \cdot b \leq -4 \cdot 10^{-306} \lor \neg \left(a \cdot a - b \cdot b \leq \infty\right):\\ \;\;\;\;\left(-b\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.9% 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 93.8%

   \[a \cdot a - b \cdot b \]
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. unpow2 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 54.6

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

5. Applied rewrites 54.6%

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

6. Taylor expanded in a around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 52.4

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

8. Applied rewrites 52.4%

   \[\leadsto \color{blue}{a \cdot a} \]
9. 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 2024337 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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