Difference of squares

Percentage Accurate: 94.1% → 96.8%

Time: 3.5s

Alternatives: 4

Speedup: 0.5×

• 43.0% 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: 94.1% 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: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{+301}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2e+301) (- (* a a) (* b b)) (* (+ a b) (+ a b))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2e+301) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = (a + b) * (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+301) then
        tmp = (a * a) - (b * b)
    else
        tmp = (a + b) * (a + b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2e+301) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = (a + b) * (a + b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 2e+301:
		tmp = (a * a) - (b * b)
	else:
		tmp = (a + b) * (a + b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 2e+301)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64(Float64(a + b) * Float64(a + b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 2e+301)
		tmp = (a * a) - (b * b);
	else
		tmp = (a + b) * (a + b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 2e+301], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 2.00000000000000011e301

   1. Initial program 100.0%

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

   if 2.00000000000000011e301 < (*.f64 a a)

   1. Initial program 78.7%

      \[a \cdot a - b \cdot b \]
   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 52.0%

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

      4. sqrt-unprod 89.3%

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

      5. sqr-neg 89.3%

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

      6. sqrt-prod 42.7%

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

      7. add-sqr-sqrt 90.7%

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

   4. Applied egg-rr 90.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2 \cdot 10^{+301}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 96.5%

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

3. Simplified 96.5%

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

5. Final simplification 96.5%

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

Alternative 3: 65.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.12 \cdot 10^{+49}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.12e+49) (* b (- b)) (* (+ a b) (+ a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.12e+49) {
		tmp = b * -b;
	} else {
		tmp = (a + b) * (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 <= 1.12d+49) then
        tmp = b * -b
    else
        tmp = (a + b) * (a + b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.12e+49) {
		tmp = b * -b;
	} else {
		tmp = (a + b) * (a + b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.12e+49:
		tmp = b * -b
	else:
		tmp = (a + b) * (a + b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.12e+49)
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(Float64(a + b) * Float64(a + b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.12e+49)
		tmp = b * -b;
	else
		tmp = (a + b) * (a + b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.12e+49], N[(b * (-b)), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.12000000000000005e49

   1. Initial program 95.0%

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

   3. Taylor expanded in a around 0 66.0%

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

      1. mul-1-neg 66.0%

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

   5. Simplified 66.0%

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

      1. unpow2 66.0%

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

   7. Applied egg-rr 66.0%

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

   if 1.12000000000000005e49 < 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 57.4%

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

      4. sqrt-unprod 94.4%

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

      5. sqr-neg 94.4%

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

      6. sqrt-prod 37.0%

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

      7. add-sqr-sqrt 87.1%

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

   4. Applied egg-rr 87.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.12 \cdot 10^{+49}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

3. Taylor expanded in a around 0 54.8%

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

   1. mul-1-neg 54.8%

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

5. Simplified 54.8%

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

   1. unpow2 54.8%

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

7. Applied egg-rr 54.8%

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

8. Final simplification 54.8%

   \[\leadsto b \cdot \left(-b\right) \]
9. 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 2024080 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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