Difference of squares

Percentage Accurate: 92.9% → 100.0%

Time: 2.1s

Alternatives: 3

Speedup: 1.0×

• 44.3% 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: 92.9% 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

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

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 93.0%

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

   1. difference-of-squares 100.0%

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

   2. *-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \left(a - b\right) \cdot \left(a + b\right) \]

Alternative 2: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4.6 \cdot 10^{-134} \lor \neg \left(a \cdot a \leq 2.2 \cdot 10^{-50}\right) \land a \cdot a \leq 350000000:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (* a a) 4.6e-134)
         (and (not (<= (* a a) 2.2e-50)) (<= (* a a) 350000000.0)))
   (* b (- b))
   (* a a)))

C

double code(double a, double b) {
	double tmp;
	if (((a * a) <= 4.6e-134) || (!((a * a) <= 2.2e-50) && ((a * a) <= 350000000.0))) {
		tmp = b * -b;
	} else {
		tmp = a * a;
	}
	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) <= 4.6d-134) .or. (.not. ((a * a) <= 2.2d-50)) .and. ((a * a) <= 350000000.0d0)) then
        tmp = b * -b
    else
        tmp = a * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((a * a) <= 4.6e-134) || (!((a * a) <= 2.2e-50) && ((a * a) <= 350000000.0))) {
		tmp = b * -b;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((a * a) <= 4.6e-134) or (not ((a * a) <= 2.2e-50) and ((a * a) <= 350000000.0)):
		tmp = b * -b
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((Float64(a * a) <= 4.6e-134) || (!(Float64(a * a) <= 2.2e-50) && (Float64(a * a) <= 350000000.0)))
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((a * a) <= 4.6e-134) || (~(((a * a) <= 2.2e-50)) && ((a * a) <= 350000000.0)))
		tmp = b * -b;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[(a * a), $MachinePrecision], 4.6e-134], And[N[Not[LessEqual[N[(a * a), $MachinePrecision], 2.2e-50]], $MachinePrecision], LessEqual[N[(a * a), $MachinePrecision], 350000000.0]]], N[(b * (-b)), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.6000000000000001e-134 or 2.1999999999999999e-50 < (*.f64 a a) < 3.5e8

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around 0 88.4%

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

      1. mul-1-neg 88.4%

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

      2. unpow2 88.4%

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

      3. distribute-rgt-neg-out 88.4%

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

   4. Simplified 88.4%

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

   if 4.6000000000000001e-134 < (*.f64 a a) < 2.1999999999999999e-50 or 3.5e8 < (*.f64 a a)

   1. Initial program 87.0%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around inf 79.4%

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

      1. unpow2 79.4%

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

   4. Simplified 79.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4.6 \cdot 10^{-134} \lor \neg \left(a \cdot a \leq 2.2 \cdot 10^{-50}\right) \land a \cdot a \leq 350000000:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 3: 53.6% 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 93.0%

   \[a \cdot a - b \cdot b \]

2. Taylor expanded in a around inf 52.6%

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

   1. unpow2 52.6%

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

4. Simplified 52.6%

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

5. Final simplification 52.6%

   \[\leadsto a \cdot a \]

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 2023297 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

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