bug366, discussion (missed optimization)

Average Error: 32.3 → 0.5

Time: 2.9s

Precision: binary64

Cost: 708

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1.3312098895913875 \cdot 10^{-279}:\\ \;\;\;\;\frac{0.5 \cdot b}{\frac{a}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -1.3312098895913875e-279) (- (/ (* 0.5 b) (/ a b)) 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 <= -1.3312098895913875e-279) {
		tmp = ((0.5 * b) / (a / b)) - 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 <= (-1.3312098895913875d-279)) then
        tmp = ((0.5d0 * b) / (a / b)) - 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 <= -1.3312098895913875e-279) {
		tmp = ((0.5 * b) / (a / b)) - 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 <= -1.3312098895913875e-279:
		tmp = ((0.5 * b) / (a / b)) - 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 <= -1.3312098895913875e-279)
		tmp = Float64(Float64(Float64(0.5 * b) / Float64(a / b)) - 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 <= -1.3312098895913875e-279)
		tmp = ((0.5 * b) / (a / b)) - 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, -1.3312098895913875e-279], N[(N[(N[(0.5 * b), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], a]

Target

Original	32.3
Target	0.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if a < -1.33120988959138753e-279

   1. Initial program 32.2

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

   2. Taylor expanded in a around -inf 4.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a} \]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\frac{0.5 \cdot b}{\frac{a}{b}} - a} \]
      Proof
      (-.f64 (/.f64 (*.f64 1/2 b) (/.f64 a b)) a): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 1/2 b) b) a)) a): 15 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 b b))) a) a): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (*.f64 b b) a))) a): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) a)) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 1/2 (/.f64 (pow.f64 b 2) a)) (neg.f64 a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (pow.f64 b 2) a)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 a))): 0 points increase in error, 0 points decrease in error

   if -1.33120988959138753e-279 < a

   1. Initial program 32.4

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

   2. Taylor expanded in a around inf 0.7

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3312098895913875 \cdot 10^{-279}:\\ \;\;\;\;\frac{0.5 \cdot b}{\frac{a}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -1.3312098895913875 \cdot 10^{-279}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 2
Error	32.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022337 
(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))))