bug366, discussion (missed optimization)

Average Error: 32.3 → 0.8

Time: 5.6s

Precision: binary64

Cost: 260

Math

\[\sqrt{a \cdot a - b \cdot b} \]

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -2.0452817502594303 \cdot 10^{-252}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

FPCore

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

↓

(FPCore (a b) :precision binary64 (if (<= a -2.0452817502594303e-252) (- a) a))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (a <= -2.0452817502594303e-252) {
		tmp = -a;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.0452817502594303d-252)) then
        tmp = -a
    else
        tmp = a
    end if
    code = tmp
end function

Java

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

↓

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.0452817502594303e-252) {
		tmp = -a;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

↓

def code(a, b):
	tmp = 0
	if a <= -2.0452817502594303e-252:
		tmp = -a
	else:
		tmp = a
	return tmp

Julia

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

↓

function code(a, b)
	tmp = 0.0
	if (a <= -2.0452817502594303e-252)
		tmp = Float64(-a);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.0452817502594303e-252)
		tmp = -a;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_] := If[LessEqual[a, -2.0452817502594303e-252], (-a), a]

Target

Original	32.3
Target	0.5
Herbie	0.8

\[\sqrt{\left|a\right| + \left|b\right|} \cdot \sqrt{\left|a\right| - \left|b\right|} \]

Derivation

1. Split input into 2 regimes

2. if a < -2.04528175025943033e-252

   1. Initial program 32.3

      \[\sqrt{a \cdot a - b \cdot b} \]

   2. Taylor expanded in a around -inf 0.6

      \[\leadsto \color{blue}{-1 \cdot a} \]

   3. Simplified 0.6

      \[\leadsto \color{blue}{-a} \]
      Proof

      [Start] 0.6	\[ -1 \cdot a \]
      rational.json-simplify-2 [=>] 0.6	\[ \color{blue}{a \cdot -1} \]
      rational.json-simplify-8 [<=] 0.6	\[ \color{blue}{-a} \]

   if -2.04528175025943033e-252 < a

   1. Initial program 32.3

      \[\sqrt{a \cdot a - b \cdot b} \]

   2. Taylor expanded in a around inf 1.1

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.0452817502594303 \cdot 10^{-252}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternatives

Alternative 1
Error	30.9
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023077 
(FPCore (a b)
  :name "bug366, discussion (missed optimization)"
  :precision binary64

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

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