Difference of squares

Percentage Accurate: 93.9% → 98.4%

Time: 3.1s

Alternatives: 4

Speedup: 0.4×

• 44.2% 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: 93.9% 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: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.85 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, a\_m, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(a\_m + b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1.85e+253) (fma a_m a_m (* b (- b))) (* (+ a_m b) (+ a_m b))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.85e+253) {
		tmp = fma(a_m, a_m, (b * -b));
	} else {
		tmp = (a_m + b) * (a_m + b);
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1.85e+253)
		tmp = fma(a_m, a_m, Float64(b * Float64(-b)));
	else
		tmp = Float64(Float64(a_m + b) * Float64(a_m + b));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1.85e+253], N[(a$95$m * a$95$m + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.85000000000000014e253

   1. Initial program 92.2%

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

      1. sqr-neg 92.2%

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

      2. cancel-sign-sub 92.2%

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

      3. fma-define 97.5%

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

   3. Simplified 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)} \]
   4. Add Preprocessing

   if 1.85000000000000014e253 < a

   1. Initial program 66.7%

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

      1. difference-of-squares 100.0%

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

      2. sub-neg 100.0%

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

      3. add-sqr-sqrt 33.3%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-prod 66.7%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+271}:\\ \;\;\;\;a\_m \cdot a\_m - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;-{b}^{2}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* b b) 1e+271) (- (* a_m a_m) (* b b)) (- (pow b 2.0))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((b * b) <= 1e+271) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = -pow(b, 2.0);
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+271) then
        tmp = (a_m * a_m) - (b * b)
    else
        tmp = -(b ** 2.0d0)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if ((b * b) <= 1e+271) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = -Math.pow(b, 2.0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (b * b) <= 1e+271:
		tmp = (a_m * a_m) - (b * b)
	else:
		tmp = -math.pow(b, 2.0)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+271)
		tmp = Float64(Float64(a_m * a_m) - Float64(b * b));
	else
		tmp = Float64(-(b ^ 2.0));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((b * b) <= 1e+271)
		tmp = (a_m * a_m) - (b * b);
	else
		tmp = -(b ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+271], N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], (-N[Power[b, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999953e270

   1. Initial program 100.0%

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

   if 9.99999999999999953e270 < (*.f64 b b)

   1. Initial program 67.6%

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

   3. Taylor expanded in a around 0 85.9%

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

      1. mul-1-neg 85.9%

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

   5. Simplified 85.9%

      \[\leadsto \color{blue}{-{b}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+271}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;-{b}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := a\_m \cdot a\_m - b \cdot b\\ \mathbf{if}\;t\_0 \leq 10^{+257}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(a\_m + b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (let* ((t_0 (- (* a_m a_m) (* b b))))
   (if (<= t_0 1e+257) t_0 (* (+ a_m b) (+ a_m b)))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double t_0 = (a_m * a_m) - (b * b);
	double tmp;
	if (t_0 <= 1e+257) {
		tmp = t_0;
	} else {
		tmp = (a_m + b) * (a_m + b);
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a_m * a_m) - (b * b)
    if (t_0 <= 1d+257) then
        tmp = t_0
    else
        tmp = (a_m + b) * (a_m + b)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double t_0 = (a_m * a_m) - (b * b);
	double tmp;
	if (t_0 <= 1e+257) {
		tmp = t_0;
	} else {
		tmp = (a_m + b) * (a_m + b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	t_0 = (a_m * a_m) - (b * b)
	tmp = 0
	if t_0 <= 1e+257:
		tmp = t_0
	else:
		tmp = (a_m + b) * (a_m + b)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	t_0 = Float64(Float64(a_m * a_m) - Float64(b * b))
	tmp = 0.0
	if (t_0 <= 1e+257)
		tmp = t_0;
	else
		tmp = Float64(Float64(a_m + b) * Float64(a_m + b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	t_0 = (a_m * a_m) - (b * b);
	tmp = 0.0;
	if (t_0 <= 1e+257)
		tmp = t_0;
	else
		tmp = (a_m + b) * (a_m + b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := Block[{t$95$0 = N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+257], t$95$0, N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 a a) (*.f64 b b)) < 1.00000000000000003e257

   1. Initial program 100.0%

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

   if 1.00000000000000003e257 < (-.f64 (*.f64 a a) (*.f64 b b))

   1. Initial program 70.1%

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

      1. difference-of-squares 100.0%

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

      2. sub-neg 100.0%

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

      3. add-sqr-sqrt 48.1%

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

      4. sqrt-unprod 88.3%

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

      5. sqr-neg 88.3%

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

      6. sqrt-prod 44.2%

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

      7. add-sqr-sqrt 83.1%

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

   4. Applied egg-rr 83.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a - b \cdot b \leq 10^{+257}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \left(a\_m + b\right) \cdot \left(a\_m + b\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 (* (+ a_m b) (+ a_m b)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	return (a_m + b) * (a_m + b);
}

Fortran

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

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return (a_m + b) * (a_m + b);
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return (a_m + b) * (a_m + b)

Julia

a_m = abs(a)
function code(a_m, b)
	return Float64(Float64(a_m + b) * Float64(a_m + b))
end

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

   1. difference-of-squares 100.0%

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

   2. sub-neg 100.0%

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

   3. add-sqr-sqrt 49.9%

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

   4. sqrt-unprod 77.5%

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

   5. sqr-neg 77.5%

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

   6. sqrt-prod 28.7%

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

   7. add-sqr-sqrt 54.4%

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

4. Applied egg-rr 54.4%

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

5. Final simplification 54.4%

   \[\leadsto \left(a + b\right) \cdot \left(a + b\right) \]
6. 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 2024071 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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