Difference of squares

Percentage Accurate: 93.9% → 100.0%

Time: 6.0s

Alternatives: 4

Speedup: 1.0×

• 42.9% 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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(b + a\right) \cdot \left(a - b\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (+ b a) (- a b)))

C

double code(double a, double b) {
	return (b + a) * (a - b);
}

Fortran

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

Java

public static double code(double a, double b) {
	return (b + a) * (a - b);
}

Python

def code(a, b):
	return (b + a) * (a - b)

Julia

function code(a, b)
	return Float64(Float64(b + a) * Float64(a - b))
end

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

   1. pow2 93.7%

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

   2. add-cbrt-cube 82.6%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\left(a \cdot a\right) \cdot a}\right)}}^{2} - b \cdot b \]

   3. pow1/3 63.9%

      \[\leadsto {\color{blue}{\left({\left(\left(a \cdot a\right) \cdot a\right)}^{0.3333333333333333}\right)}}^{2} - b \cdot b \]

   4. pow-pow 63.9%

      \[\leadsto \color{blue}{{\left(\left(a \cdot a\right) \cdot a\right)}^{\left(0.3333333333333333 \cdot 2\right)}} - b \cdot b \]

   5. pow3 63.9%

      \[\leadsto {\color{blue}{\left({a}^{3}\right)}}^{\left(0.3333333333333333 \cdot 2\right)} - b \cdot b \]

   6. metadata-eval 63.9%

      \[\leadsto {\left({a}^{3}\right)}^{\color{blue}{0.6666666666666666}} - b \cdot b \]

4. Applied egg-rr 63.9%

   \[\leadsto \color{blue}{{\left({a}^{3}\right)}^{0.6666666666666666}} - b \cdot b \]
5. Step-by-step derivation

   1. pow-pow 93.7%

      \[\leadsto \color{blue}{{a}^{\left(3 \cdot 0.6666666666666666\right)}} - b \cdot b \]

   2. metadata-eval 93.7%

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

   3. pow2 93.7%

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

   4. difference-of-squares 100.0%

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

   5. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{-69}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2e-69) (* b (- b)) (* a (- a b))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2e-69) {
		tmp = b * -b;
	} else {
		tmp = a * (a - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 2d-69) then
        tmp = b * -b
    else
        tmp = a * (a - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2e-69) {
		tmp = b * -b;
	} else {
		tmp = a * (a - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 2e-69:
		tmp = b * -b
	else:
		tmp = a * (a - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 2e-69)
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(a * Float64(a - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 2e-69)
		tmp = b * -b;
	else
		tmp = a * (a - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 2e-69], N[(b * (-b)), $MachinePrecision], N[(a * N[(a - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1.9999999999999999e-69

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 86.0%

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

      1. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

      1. pow2 86.0%

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

      2. distribute-lft-neg-in 86.0%

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

   7. Applied egg-rr 86.0%

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

   if 1.9999999999999999e-69 < (*.f64 a a)

   1. Initial program 88.6%

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

      1. pow2 88.6%

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

      2. add-cbrt-cube 72.2%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{\left(a \cdot a\right) \cdot a}\right)}}^{2} - b \cdot b \]

      3. pow1/3 49.1%

         \[\leadsto {\color{blue}{\left({\left(\left(a \cdot a\right) \cdot a\right)}^{0.3333333333333333}\right)}}^{2} - b \cdot b \]

      4. pow-pow 49.1%

         \[\leadsto \color{blue}{{\left(\left(a \cdot a\right) \cdot a\right)}^{\left(0.3333333333333333 \cdot 2\right)}} - b \cdot b \]

      5. pow3 49.1%

         \[\leadsto {\color{blue}{\left({a}^{3}\right)}}^{\left(0.3333333333333333 \cdot 2\right)} - b \cdot b \]

      6. metadata-eval 49.1%

         \[\leadsto {\left({a}^{3}\right)}^{\color{blue}{0.6666666666666666}} - b \cdot b \]

   4. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{{\left({a}^{3}\right)}^{0.6666666666666666}} - b \cdot b \]
   5. Step-by-step derivation

      1. pow-pow 88.6%

         \[\leadsto \color{blue}{{a}^{\left(3 \cdot 0.6666666666666666\right)}} - b \cdot b \]

      2. metadata-eval 88.6%

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

      3. pow2 88.6%

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

      4. difference-of-squares 100.0%

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

      5. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in b around 0 77.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{-69}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-114}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 2e-114) (* a a) (* b (- b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-114) {
		tmp = a * a;
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d-114) then
        tmp = a * a
    else
        tmp = b * -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-114) {
		tmp = a * a;
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-114:
		tmp = a * a
	else:
		tmp = b * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-114)
		tmp = Float64(a * a);
	else
		tmp = Float64(b * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-114)
		tmp = a * a;
	else
		tmp = b * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-114], N[(a * a), $MachinePrecision], N[(b * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e-114

   1. Initial program 99.9%

      \[a \cdot a - b \cdot b \]
   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 51.8%

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

      4. sqrt-unprod 96.9%

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

      5. sqr-neg 96.9%

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

      6. sqrt-prod 45.0%

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

      7. add-sqr-sqrt 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in a around inf 88.3%

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

   6. Taylor expanded in a around inf 88.5%

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

   if 2.0000000000000001e-114 < (*.f64 b b)

   1. Initial program 89.3%

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

   3. Taylor expanded in a around 0 75.6%

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

      1. neg-mul-1 75.6%

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

   5. Simplified 75.6%

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

      1. pow2 75.6%

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

      2. distribute-lft-neg-in 75.6%

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

   7. Applied egg-rr 75.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-114}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.0% 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 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 45.6%

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

   4. sqrt-unprod 71.1%

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

   5. sqr-neg 71.1%

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

   6. sqrt-prod 26.6%

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

   7. add-sqr-sqrt 50.4%

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

4. Applied egg-rr 50.4%

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

5. Taylor expanded in a around inf 56.1%

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

6. Taylor expanded in a around inf 51.1%

   \[\leadsto \color{blue}{a} \cdot a \]
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 2024172 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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