Difference of squares

Percentage Accurate: 93.9% → 97.4%

Time: 2.0s

Alternatives: 4

Speedup: 0.8×

• 42.8% 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: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.3e+191) (fma a a (* b (- b))) (* a a)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.3e+191) {
		tmp = fma(a, a, (b * -b));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.3e+191)
		tmp = fma(a, a, Float64(b * Float64(-b)));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.3e+191], N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.2999999999999999e191

   1. Initial program 96.0%

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

      1. sqr-neg 96.0%

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

      2. cancel-sign-sub 96.0%

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

      3. fma-def 98.7%

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

   3. Simplified 98.7%

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

   if 2.2999999999999999e191 < a

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 2: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.35e+154) (- (* a a) (* b b)) (* a a)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.35e+154) {
		tmp = (a * a) - (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 <= 1.35d+154) then
        tmp = (a * a) - (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 <= 1.35e+154) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.35e+154:
		tmp = (a * a) - (b * b)
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.35e+154)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.35e+154)
		tmp = (a * a) - (b * b);
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.35e+154], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.35000000000000003e154

   1. Initial program 96.0%

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

   if 1.35000000000000003e154 < a

   1. Initial program 74.2%

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

   2. Taylor expanded in a around inf 96.8%

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

      1. unpow2 96.8%

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

   4. Simplified 96.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 3: 66.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 2.2e-40) (* a a) (* b (- b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-40) {
		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 <= 2.2d-40) 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 <= 2.2e-40) {
		tmp = a * a;
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.2e-40:
		tmp = a * a
	else:
		tmp = b * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.2e-40)
		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 <= 2.2e-40)
		tmp = a * a;
	else
		tmp = b * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.2e-40], N[(a * a), $MachinePrecision], N[(b * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.20000000000000009e-40

   1. Initial program 94.1%

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

   2. Taylor expanded in a around inf 65.2%

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

      1. unpow2 65.2%

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

   4. Simplified 65.2%

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

   if 2.20000000000000009e-40 < b

   1. Initial program 91.2%

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

   2. Taylor expanded in a around 0 79.5%

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

      1. unpow2 79.5%

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

      2. mul-1-neg 79.5%

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

      3. distribute-rgt-neg-in 79.5%

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

   4. Simplified 79.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]

Alternative 4: 54.5% 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 93.3%

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

2. Taylor expanded in a around inf 53.6%

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

   1. unpow2 53.6%

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

4. Simplified 53.6%

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

5. Final simplification 53.6%

   \[\leadsto a \cdot a \]

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 2023272 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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