Difference of squares

Percentage Accurate: 93.6% → 98.6%

Time: 5.3s

Alternatives: 4

Speedup: 0.6×

• 43.2% 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: 93.6% 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: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 3.5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, a\_m, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(a\_m + b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 3.5e+172) (fma 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 <= 3.5e+172) {
		tmp = fma(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 <= 3.5e+172)
		tmp = fma(a_m, a_m, Float64(b * Float64(-b)));
	else
		tmp = Float64(a_m * 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, 3.5e+172], N[(a$95$m * a$95$m + 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 < 3.49999999999999977e172

   1. Initial program 97.4%

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

      1. sqr-neg 97.4%

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

      2. cancel-sign-sub 97.4%

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

      3. fma-define 97.8%

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

   3. Simplified 97.8%

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

   if 3.49999999999999977e172 < a

   1. Initial program 80.8%

      \[a \cdot a - b \cdot b \]
   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 69.2%

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

      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 30.8%

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

      7. add-sqr-sqrt 92.3%

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in a around inf 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 6.8 \cdot 10^{+115}:\\ \;\;\;\;a\_m \cdot a\_m - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(a\_m + b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 6.8e+115) (- (* 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 <= 6.8e+115) {
		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 <= 6.8d+115) 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 <= 6.8e+115) {
		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 <= 6.8e+115:
		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 <= 6.8e+115)
		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 <= 6.8e+115)
		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, 6.8e+115], 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 < 6.8000000000000001e115

   1. Initial program 97.3%

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

   if 6.8000000000000001e115 < a

   1. Initial program 85.7%

      \[a \cdot a - b \cdot b \]
   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 68.6%

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

      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 31.4%

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

      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 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.8 \cdot 10^{+115}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \cdot a\_m \leq 5 \cdot 10^{+111}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* a_m a_m) 5e+111) (* 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) <= 5e+111) {
		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) <= 5d+111) 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) <= 5e+111) {
		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) <= 5e+111:
		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) <= 5e+111)
		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) <= 5e+111)
		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], 5e+111], N[(b * (-b)), $MachinePrecision], N[(a$95$m * a$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.9999999999999997e111

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 80.6%

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

      1. neg-mul-1 80.6%

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

   5. Simplified 80.6%

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

      1. unpow2 80.6%

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

      2. distribute-lft-neg-in 80.6%

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

   7. Applied egg-rr 80.6%

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

   if 4.9999999999999997e111 < (*.f64 a a)

   1. Initial program 88.9%

      \[a \cdot a - b \cdot b \]
   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 55.6%

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

      4. sqrt-unprod 87.9%

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

      5. sqr-neg 87.9%

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

      6. sqrt-prod 37.4%

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

      7. add-sqr-sqrt 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in a around inf 91.1%

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

   6. Taylor expanded in a around inf 86.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+111}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.5% 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 95.7%

   \[a \cdot a - b \cdot b \]
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 53.0%

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

   4. sqrt-unprod 73.6%

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

   5. sqr-neg 73.6%

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

   6. sqrt-prod 22.4%

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

   7. add-sqr-sqrt 53.3%

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

4. Applied egg-rr 53.3%

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

5. Taylor expanded in a around inf 56.7%

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

6. Taylor expanded in a around inf 54.0%

   \[\leadsto \color{blue}{a} \cdot a \]
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 2024188 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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