bug366, discussion (missed optimization)

Average Error: 31.3 → 0.5

Time: 3.9s

Precision: binary64

Cost: 900

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1.8646666152228868 \cdot 10^{-290}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0 + \frac{b}{a} \cdot \left(-b\right)\right) + a\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8646666152228868e-290)
   (- a)
   (+ (* 0.5 (+ 0.0 (* (/ b a) (- b)))) a)))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (a <= -1.8646666152228868e-290) {
		tmp = -a;
	} else {
		tmp = (0.5 * (0.0 + ((b / a) * -b))) + 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 <= (-1.8646666152228868d-290)) then
        tmp = -a
    else
        tmp = (0.5d0 * (0.0d0 + ((b / a) * -b))) + 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 <= -1.8646666152228868e-290) {
		tmp = -a;
	} else {
		tmp = (0.5 * (0.0 + ((b / a) * -b))) + a;
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if a <= -1.8646666152228868e-290:
		tmp = -a
	else:
		tmp = (0.5 * (0.0 + ((b / a) * -b))) + 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 <= -1.8646666152228868e-290)
		tmp = Float64(-a);
	else
		tmp = Float64(Float64(0.5 * Float64(0.0 + Float64(Float64(b / a) * Float64(-b)))) + 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 <= -1.8646666152228868e-290)
		tmp = -a;
	else
		tmp = (0.5 * (0.0 + ((b / a) * -b))) + 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, -1.8646666152228868e-290], (-a), N[(N[(0.5 * N[(0.0 + N[(N[(b / a), $MachinePrecision] * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Target

Original	31.3
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 < -1.86466661522288677e-290

   1. Initial program 31.1

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

   2. Simplified 31.1

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

   3. Taylor expanded in a around -inf 0.7

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

   4. Simplified 0.7

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

   if -1.86466661522288677e-290 < a

   1. Initial program 31.5

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

   2. Simplified 31.5

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

   3. Taylor expanded in a around inf 4.3

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

   4. Simplified 4.3

      \[\leadsto \color{blue}{0.5 \cdot \left(0 + \frac{\left(-b \cdot b\right) - \left(0 \cdot 0\right) \cdot 0.25}{a}\right) + a} \]
      Proof

   5. Applied egg-rr 0.3

      \[\leadsto 0.5 \cdot \left(0 + \color{blue}{\frac{b}{a} \cdot \left(-b\right)}\right) + a \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	0.7
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -1.8646666152228868 \cdot 10^{-290}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 2
Error	32.4
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023010 
(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))))