Difference of squares

Percentage Accurate: 93.6% → 98.7%

Time: 4.6s

Alternatives: 4

Speedup: 0.6×

• 44.7% 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: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a - b \cdot b \leq -2 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} \cdot \left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (- (* a a) (* b b)) -2e-239)
   (fma a a (* b (- b)))
   (* (pow a 2.0) (- 1.0 (* (/ b a) (/ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (((a * a) - (b * b)) <= -2e-239) {
		tmp = fma(a, a, (b * -b));
	} else {
		tmp = pow(a, 2.0) * (1.0 - ((b / a) * (b / a)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64(a * a) - Float64(b * b)) <= -2e-239)
		tmp = fma(a, a, Float64(b * Float64(-b)));
	else
		tmp = Float64((a ^ 2.0) * Float64(1.0 - Float64(Float64(b / a) * Float64(b / a))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], -2e-239], N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(b / a), $MachinePrecision] * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 a a) (*.f64 b b)) < -2.0000000000000002e-239

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if -2.0000000000000002e-239 < (-.f64 (*.f64 a a) (*.f64 b b))

   1. Initial program 83.2%

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

   3. Taylor expanded in a around inf 73.8%

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(1 + -1 \cdot \frac{{b}^{2}}{{a}^{2}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 73.8%

         \[\leadsto {a}^{2} \cdot \left(1 + \color{blue}{\left(-\frac{{b}^{2}}{{a}^{2}}\right)}\right) \]

      2. unsub-neg 73.8%

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(1 - \frac{{b}^{2}}{{a}^{2}}\right)} \]

   5. Simplified 73.8%

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(1 - \frac{{b}^{2}}{{a}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 73.8%

         \[\leadsto {a}^{2} \cdot \left(1 - \frac{\color{blue}{b \cdot b}}{{a}^{2}}\right) \]

      2. unpow2 73.8%

         \[\leadsto {a}^{2} \cdot \left(1 - \frac{b \cdot b}{\color{blue}{a \cdot a}}\right) \]

      3. times-frac 100.0%

         \[\leadsto {a}^{2} \cdot \left(1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {a}^{2} \cdot \left(1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a - b \cdot b \leq -2 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} \cdot \left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 4.1e+250) (fma a a (* b (- b))) (* (+ a b) (+ a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 4.1e+250) {
		tmp = fma(a, a, (b * -b));
	} else {
		tmp = (a + b) * (a + b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 4.1e+250)
		tmp = fma(a, a, Float64(b * Float64(-b)));
	else
		tmp = Float64(Float64(a + b) * Float64(a + b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 4.1e+250], N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.09999999999999999e250

   1. Initial program 91.7%

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

      1. sqr-neg 91.7%

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

      2. cancel-sign-sub 91.7%

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

      3. fma-define 96.3%

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

   3. Simplified 96.3%

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

   if 4.09999999999999999e250 < a

   1. Initial program 64.3%

      \[a \cdot a - b \cdot b \]
   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 92.9%

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

      5. sqr-neg 92.9%

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

      6. sqrt-prod 42.9%

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

      7. add-sqr-sqrt 92.9%

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

   4. Applied egg-rr 92.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+132}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 8e+132) (- (* a a) (* b b)) (* (+ a b) (+ a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 8e+132) {
		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 <= 8d+132) 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 <= 8e+132) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = (a + b) * (a + b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 8e+132:
		tmp = (a * a) - (b * b)
	else:
		tmp = (a + b) * (a + b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 8e+132)
		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 <= 8e+132)
		tmp = (a * a) - (b * b);
	else
		tmp = (a + b) * (a + b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 8e+132], 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 a < 7.99999999999999993e132

   1. Initial program 94.4%

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

   if 7.99999999999999993e132 < a

   1. Initial program 69.8%

      \[a \cdot a - b \cdot b \]
   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 58.1%

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

      4. sqrt-unprod 93.0%

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

      5. sqr-neg 93.0%

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

      6. sqrt-prod 34.9%

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

      7. add-sqr-sqrt 83.7%

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

   4. Applied egg-rr 83.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+132}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.1% 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]

Derivation

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 47.2%

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

   4. sqrt-unprod 72.2%

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

   5. sqr-neg 72.2%

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

   6. sqrt-prod 27.7%

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

   7. add-sqr-sqrt 52.8%

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

4. Applied egg-rr 52.8%

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

5. Final simplification 52.8%

   \[\leadsto \left(a + b\right) \cdot \left(a + b\right) \]
6. 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 2024085 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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