Difference of squares

Percentage Accurate: 93.4% → 98.5%

Time: 4.3s

Alternatives: 4

Speedup: 0.5×

• 43.8% 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: 93.4% 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: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, a\_m, -b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1e+221) (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 <= 1e+221) {
		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 <= 1e+221)
		tmp = fma(a_m, a_m, Float64(-Float64(b * 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, 1e+221], 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 < 1e221

   1. Initial program 93.8%

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

      1. sqr-neg 93.8%

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

      2. cancel-sign-sub 93.8%

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

      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 1e221 < a

   1. Initial program 92.9%

      \[a \cdot a - b \cdot b \]
   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 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 100.0%

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

   4. Applied egg-rr 100.0%

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

   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 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(a, a, -b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+303}:\\ \;\;\;\;a\_m \cdot a\_m - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;-b \cdot b\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* b b) 1e+303) (- (* a_m a_m) (* b b)) (- (* b b))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = -(b * 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 ((b * b) <= 1d+303) then
        tmp = (a_m * a_m) - (b * b)
    else
        tmp = -(b * 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 ((b * b) <= 1e+303) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = -(b * b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (b * b) <= 1e+303:
		tmp = (a_m * a_m) - (b * b)
	else:
		tmp = -(b * b)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+303)
		tmp = Float64(Float64(a_m * a_m) - Float64(b * b));
	else
		tmp = Float64(-Float64(b * b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((b * b) <= 1e+303)
		tmp = (a_m * a_m) - (b * b);
	else
		tmp = -(b * b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+303], N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], (-N[(b * b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e303

   1. Initial program 100.0%

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

   if 1e303 < (*.f64 b b)

   1. Initial program 76.1%

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

   3. Taylor expanded in a around 0 89.6%

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

      1. neg-mul-1 89.6%

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

   5. Simplified 89.6%

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

      1. unpow2 89.6%

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

      2. distribute-lft-neg-in 89.6%

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

   7. Applied egg-rr 89.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+303}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;-b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-18}:\\ \;\;\;\;a\_m \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;-b \cdot b\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* b b) 1e-18) (* a_m a_m) (- (* b b))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((b * b) <= 1e-18) {
		tmp = a_m * a_m;
	} else {
		tmp = -(b * 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 ((b * b) <= 1d-18) then
        tmp = a_m * a_m
    else
        tmp = -(b * 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 ((b * b) <= 1e-18) {
		tmp = a_m * a_m;
	} else {
		tmp = -(b * b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (b * b) <= 1e-18:
		tmp = a_m * a_m
	else:
		tmp = -(b * b)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-18)
		tmp = Float64(a_m * a_m);
	else
		tmp = Float64(-Float64(b * b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((b * b) <= 1e-18)
		tmp = a_m * a_m;
	else
		tmp = -(b * b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-18], N[(a$95$m * a$95$m), $MachinePrecision], (-N[(b * b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.0000000000000001e-18

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]
   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 47.1%

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

      4. sqrt-unprod 94.5%

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

      5. sqr-neg 94.5%

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

      6. sqrt-prod 47.3%

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

      7. add-sqr-sqrt 85.8%

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

   4. Applied egg-rr 85.8%

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

   5. Taylor expanded in a around inf 86.0%

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

   6. Taylor expanded in a around inf 86.0%

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

   if 1.0000000000000001e-18 < (*.f64 b b)

   1. Initial program 87.9%

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

   3. Taylor expanded in a around 0 76.0%

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

      1. neg-mul-1 76.0%

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

   5. Simplified 76.0%

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

      1. unpow2 76.0%

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

      2. distribute-lft-neg-in 76.0%

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

   7. Applied egg-rr 76.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-18}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;-b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.7% 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 93.7%

   \[a \cdot a - b \cdot b \]
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 50.3%

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

   4. sqrt-unprod 76.8%

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

   5. sqr-neg 76.8%

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

   6. sqrt-prod 27.9%

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

   7. add-sqr-sqrt 53.1%

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

4. Applied egg-rr 53.1%

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

5. Taylor expanded in a around inf 56.8%

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

6. Taylor expanded in a around inf 53.9%

   \[\leadsto \color{blue}{a} \cdot a \]
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 2024133 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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