Difference of squares

Percentage Accurate: 93.3% → 97.9%

Time: 4.2s

Alternatives: 4

Speedup: 0.6×

• 44.1% 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: 93.3% 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: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 5.2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, a\_m, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 5.2e+179) (fma a_m a_m (* b (- b))) (* a_m a_m)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 5.2e+179) {
		tmp = fma(a_m, a_m, (b * -b));
	} else {
		tmp = a_m * a_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 5.2e+179)
		tmp = fma(a_m, a_m, Float64(b * Float64(-b)));
	else
		tmp = Float64(a_m * a_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 5.2e+179], N[(a$95$m * a$95$m + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(a$95$m * a$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.2000000000000004e179

   1. Initial program 96.1%

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

      1. sqr-neg 96.1%

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

      2. cancel-sign-sub 96.1%

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

      3. fma-define 97.4%

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

   3. Simplified 97.4%

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

   if 5.2000000000000004e179 < a

   1. Initial program 82.1%

      \[a \cdot a - b \cdot b \]
   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 50.0%

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

      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 50.0%

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

      7. add-sqr-sqrt 96.4%

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

   4. Applied egg-rr 96.4%

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

   5. Taylor expanded in a around inf 100.0%

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

   6. Taylor expanded in a around inf 96.4%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;a\_m \cdot a\_m - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(a\_m + b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 3.8e+147) (- (* a_m a_m) (* b b)) (* a_m (+ a_m b))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 3.8e+147) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = a_m * (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) :: tmp
    if (a_m <= 3.8d+147) then
        tmp = (a_m * a_m) - (b * b)
    else
        tmp = a_m * (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 tmp;
	if (a_m <= 3.8e+147) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = a_m * (a_m + b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if a_m <= 3.8e+147:
		tmp = (a_m * a_m) - (b * b)
	else:
		tmp = a_m * (a_m + b)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 3.8e+147)
		tmp = Float64(Float64(a_m * a_m) - Float64(b * b));
	else
		tmp = Float64(a_m * Float64(a_m + b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if (a_m <= 3.8e+147)
		tmp = (a_m * a_m) - (b * b);
	else
		tmp = a_m * (a_m + b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 3.8e+147], N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.7999999999999997e147

   1. Initial program 96.4%

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

   if 3.7999999999999997e147 < a

   1. Initial program 82.9%

      \[a \cdot a - b \cdot b \]
   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 48.6%

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

      4. sqrt-unprod 97.1%

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

      5. sqr-neg 97.1%

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

      6. sqrt-prod 48.6%

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

      7. add-sqr-sqrt 94.3%

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in a around inf 97.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \cdot a\_m \leq 2 \cdot 10^{-76}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* a_m a_m) 2e-76) (* b (- b)) (* a_m a_m)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 2e-76) {
		tmp = b * -b;
	} else {
		tmp = a_m * a_m;
	}
	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 ((a_m * a_m) <= 2d-76) then
        tmp = b * -b
    else
        tmp = a_m * a_m
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 2e-76) {
		tmp = b * -b;
	} else {
		tmp = a_m * a_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (a_m * a_m) <= 2e-76:
		tmp = b * -b
	else:
		tmp = a_m * a_m
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (Float64(a_m * a_m) <= 2e-76)
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(a_m * a_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((a_m * a_m) <= 2e-76)
		tmp = b * -b;
	else
		tmp = a_m * a_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 2e-76], N[(b * (-b)), $MachinePrecision], N[(a$95$m * a$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1.99999999999999985e-76

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 85.8%

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

      1. neg-mul-1 85.8%

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

   5. Simplified 85.8%

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

      1. unpow2 85.8%

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

      2. distribute-lft-neg-in 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 1.99999999999999985e-76 < (*.f64 a a)

   1. Initial program 90.2%

      \[a \cdot a - b \cdot b \]
   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 48.2%

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

      4. sqrt-unprod 85.7%

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

      5. sqr-neg 85.7%

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

      6. sqrt-prod 41.7%

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

      7. add-sqr-sqrt 81.5%

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

   4. Applied egg-rr 81.5%

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

   5. Taylor expanded in a around inf 87.1%

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

   6. Taylor expanded in a around inf 82.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{-76}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 * a_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

   \[a \cdot a - b \cdot b \]
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 45.6%

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

   4. sqrt-unprod 71.9%

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

   5. sqr-neg 71.9%

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

   6. sqrt-prod 28.5%

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

   7. add-sqr-sqrt 57.9%

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

4. Applied egg-rr 57.9%

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

5. Taylor expanded in a around inf 61.6%

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

6. Taylor expanded in a around inf 58.7%

   \[\leadsto \color{blue}{a} \cdot a \]
7. 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 2024185 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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