Difference of squares

Percentage Accurate: 93.5% → 100.0%

Time: 5.8s

Alternatives: 3

Speedup: 1.0×

• 44.8% 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: 93.5% 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, 1.0× 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 94.1%

   \[a \cdot a - b \cdot b \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -b \cdot b\\ \mathbf{if}\;a \cdot a \leq 6.5 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \cdot a \leq 1.95 \cdot 10^{+61}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;a \cdot a \leq 8.5 \cdot 10^{+168}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* b b))))
   (if (<= (* a a) 6.5e-72)
     t_0
     (if (<= (* a a) 1.95e+61)
       (* a a)
       (if (<= (* a a) 8.5e+168) t_0 (* a a))))))

C

double code(double a, double b) {
	double t_0 = -(b * b);
	double tmp;
	if ((a * a) <= 6.5e-72) {
		tmp = t_0;
	} else if ((a * a) <= 1.95e+61) {
		tmp = a * a;
	} else if ((a * a) <= 8.5e+168) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -(b * b)
    if ((a * a) <= 6.5d-72) then
        tmp = t_0
    else if ((a * a) <= 1.95d+61) then
        tmp = a * a
    else if ((a * a) <= 8.5d+168) then
        tmp = t_0
    else
        tmp = a * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = -(b * b);
	double tmp;
	if ((a * a) <= 6.5e-72) {
		tmp = t_0;
	} else if ((a * a) <= 1.95e+61) {
		tmp = a * a;
	} else if ((a * a) <= 8.5e+168) {
		tmp = t_0;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = -(b * b)
	tmp = 0
	if (a * a) <= 6.5e-72:
		tmp = t_0
	elif (a * a) <= 1.95e+61:
		tmp = a * a
	elif (a * a) <= 8.5e+168:
		tmp = t_0
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	t_0 = Float64(-Float64(b * b))
	tmp = 0.0
	if (Float64(a * a) <= 6.5e-72)
		tmp = t_0;
	elseif (Float64(a * a) <= 1.95e+61)
		tmp = Float64(a * a);
	elseif (Float64(a * a) <= 8.5e+168)
		tmp = t_0;
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = -(b * b);
	tmp = 0.0;
	if ((a * a) <= 6.5e-72)
		tmp = t_0;
	elseif ((a * a) <= 1.95e+61)
		tmp = a * a;
	elseif ((a * a) <= 8.5e+168)
		tmp = t_0;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = (-N[(b * b), $MachinePrecision])}, If[LessEqual[N[(a * a), $MachinePrecision], 6.5e-72], t$95$0, If[LessEqual[N[(a * a), $MachinePrecision], 1.95e+61], N[(a * a), $MachinePrecision], If[LessEqual[N[(a * a), $MachinePrecision], 8.5e+168], t$95$0, N[(a * a), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 6.4999999999999997e-72 or 1.94999999999999994e61 < (*.f64 a a) < 8.50000000000000069e168

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 6.4999999999999997e-72 < (*.f64 a a) < 1.94999999999999994e61 or 8.50000000000000069e168 < (*.f64 a a)

   1. Initial program 87.7%

      \[a \cdot a - b \cdot b \]

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. 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 94.1%

   \[a \cdot a - b \cdot b \]

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in a around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Developer target: 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 2024110 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :alt
  (* (+ a b) (- a b))

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