bug366, discussion (missed optimization)

Percentage Accurate: 53.7% → 98.8%

Time: 3.9s

Precision: binary64

Cost: 13380

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -5e-271) (- a) (* (sqrt (- a b)) (sqrt (+ a b)))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (a <= -5e-271) {
		tmp = -a;
	} else {
		tmp = sqrt((a - b)) * sqrt((a + b));
	}
	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 <= (-5d-271)) then
        tmp = -a
    else
        tmp = sqrt((a - b)) * sqrt((a + b))
    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 <= -5e-271) {
		tmp = -a;
	} else {
		tmp = Math.sqrt((a - b)) * Math.sqrt((a + b));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if a <= -5e-271:
		tmp = -a
	else:
		tmp = math.sqrt((a - b)) * math.sqrt((a + b))
	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 <= -5e-271)
		tmp = Float64(-a);
	else
		tmp = Float64(sqrt(Float64(a - b)) * sqrt(Float64(a + b)));
	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 <= -5e-271)
		tmp = -a;
	else
		tmp = sqrt((a - b)) * sqrt((a + b));
	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, -5e-271], (-a), N[(N[Sqrt[N[(a - b), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	53.7%
Target	99.2%
Herbie	98.8%

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

Derivation

1. Split input into 2 regimes

2. if a < -5.0000000000000002e-271

   1. Initial program 60.7%

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

   2. Simplified 60.8%

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

   3. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in a around -inf 99.9%

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

   5. Simplified 99.9%

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

   if -5.0000000000000002e-271 < a

   1. Initial program 47.9%

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

   2. Simplified 48.5%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 3
Accuracy	50.3%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023161 
(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))))