Difference of squares

Percentage Accurate: 93.6% → 98.2%
Time: 3.3s
Alternatives: 4
Speedup: 0.8×

Specification

?
\[\begin{array}{l} \\ a \cdot a - b \cdot b \end{array} \]
(FPCore (a b) :precision binary64 (- (* a a) (* b b)))
double code(double a, double b) {
	return (a * a) - (b * b);
}

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

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

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

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

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

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

\begin{array}{l}

\\
a \cdot a - b \cdot b
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 4 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 93.6% accurate, 1.0× speedup?

\[\begin{array}{l} \\ a \cdot a - b \cdot b \end{array} \]
(FPCore (a b) :precision binary64 (- (* a a) (* b b)))
double code(double a, double b) {
	return (a * a) - (b * b);
}

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

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

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

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

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

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

\begin{array}{l}

\\
a \cdot a - b \cdot b
\end{array}

Alternative 1: 98.2% accurate, 0.1× speedup?

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(a_m, a_m, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\ \end{array} \end{array} \]
a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 5e+262) (fma a_m a_m (* b (- b))) (* (- a_m b) (- a_m b))))
a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 5e+262) {
		tmp = fma(a_m, a_m, (b * -b));
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 5e+262)
		tmp = fma(a_m, a_m, Float64(b * Float64(-b)));
	else
		tmp = Float64(Float64(a_m - b) * Float64(a_m - b));
	end
	return tmp
end

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 5e+262], N[(a$95$m * a$95$m + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m - b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
a_m = \left|a\right|

\\
\begin{array}{l}
\mathbf{if}\;a_m \leq 5 \cdot 10^{+262}:\\
\;\;\;\;\mathsf{fma}\left(a_m, a_m, b \cdot \left(-b\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 76.8% accurate, 0.5× speedup?

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 1.2 \cdot 10^{-45} \lor \neg \left(a_m \leq 2 \cdot 10^{+71}\right) \land a_m \leq 8.8 \cdot 10^{+82}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\ \end{array} \end{array} \]
a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (or (<= a_m 1.2e-45) (and (not (<= a_m 2e+71)) (<= a_m 8.8e+82)))
   (* b (- b))
   (* (- a_m b) (- a_m b))))
a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((a_m <= 1.2e-45) || (!(a_m <= 2e+71) && (a_m <= 8.8e+82))) {
		tmp = b * -b;
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a_m <= 1.2d-45) .or. (.not. (a_m <= 2d+71)) .and. (a_m <= 8.8d+82)) then
        tmp = b * -b
    else
        tmp = (a_m - b) * (a_m - b)
    end if
    code = tmp
end function

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if ((a_m <= 1.2e-45) || (!(a_m <= 2e+71) && (a_m <= 8.8e+82))) {
		tmp = b * -b;
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (a_m <= 1.2e-45) or (not (a_m <= 2e+71) and (a_m <= 8.8e+82)):
		tmp = b * -b
	else:
		tmp = (a_m - b) * (a_m - b)
	return tmp

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if ((a_m <= 1.2e-45) || (!(a_m <= 2e+71) && (a_m <= 8.8e+82)))
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(Float64(a_m - b) * Float64(a_m - b));
	end
	return tmp
end

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((a_m <= 1.2e-45) || (~((a_m <= 2e+71)) && (a_m <= 8.8e+82)))
		tmp = b * -b;
	else
		tmp = (a_m - b) * (a_m - b);
	end
	tmp_2 = tmp;
end

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[Or[LessEqual[a$95$m, 1.2e-45], And[N[Not[LessEqual[a$95$m, 2e+71]], $MachinePrecision], LessEqual[a$95$m, 8.8e+82]]], N[(b * (-b)), $MachinePrecision], N[(N[(a$95$m - b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
a_m = \left|a\right|

\\
\begin{array}{l}
\mathbf{if}\;a_m \leq 1.2 \cdot 10^{-45} \lor \neg \left(a_m \leq 2 \cdot 10^{+71}\right) \land a_m \leq 8.8 \cdot 10^{+82}:\\
\;\;\;\;b \cdot \left(-b\right)\\

\mathbf{else}:\\
\;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 96.8% accurate, 0.8× speedup?

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 6 \cdot 10^{+149}:\\ \;\;\;\;a_m \cdot a_m - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\ \end{array} \end{array} \]
a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 6e+149) (- (* a_m a_m) (* b b)) (* (- a_m b) (- a_m b))))
a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 6e+149) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a_m <= 6d+149) then
        tmp = (a_m * a_m) - (b * b)
    else
        tmp = (a_m - b) * (a_m - b)
    end if
    code = tmp
end function

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if (a_m <= 6e+149) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if a_m <= 6e+149:
		tmp = (a_m * a_m) - (b * b)
	else:
		tmp = (a_m - b) * (a_m - b)
	return tmp

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 6e+149)
		tmp = Float64(Float64(a_m * a_m) - Float64(b * b));
	else
		tmp = Float64(Float64(a_m - b) * Float64(a_m - b));
	end
	return tmp
end

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if (a_m <= 6e+149)
		tmp = (a_m * a_m) - (b * b);
	else
		tmp = (a_m - b) * (a_m - b);
	end
	tmp_2 = tmp;
end

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 6e+149], N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m - b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
a_m = \left|a\right|

\\
\begin{array}{l}
\mathbf{if}\;a_m \leq 6 \cdot 10^{+149}:\\
\;\;\;\;a_m \cdot a_m - b \cdot b\\

\mathbf{else}:\\
\;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 53.2% accurate, 1.8× speedup?

\[\begin{array}{l} a_m = \left|a\right| \\ b \cdot \left(-b\right) \end{array} \]
a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 (* b (- b)))
a_m = fabs(a);
double code(double a_m, double b) {
	return b * -b;
}

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

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return b * -b;
}

a_m = math.fabs(a)
def code(a_m, b):
	return b * -b

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

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

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(b * (-b)), $MachinePrecision]

\begin{array}{l}
a_m = \left|a\right|

\\
b \cdot \left(-b\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Developer target: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left(a + b\right) \cdot \left(a - b\right) \end{array} \]
(FPCore (a b) :precision binary64 (* (+ a b) (- a b)))
double code(double a, double b) {
	return (a + b) * (a - b);
}

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

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

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

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

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

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

\begin{array}{l}

\\
\left(a + b\right) \cdot \left(a - b\right)
\end{array}

Reproduce

?
herbie shell --seed 2023347 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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