bug366, discussion (missed optimization)

Average Error: 32.0 → 0.4

Time: 4.6s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -3.901864712582398 \cdot 10^{-273}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -3.901864712582398e-273)
   (- a)
   (/ (- a b) (sqrt (/ (- a b) (+ 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 <= -3.901864712582398e-273) {
		tmp = -a;
	} else {
		tmp = (a - b) / sqrt(((a - b) / (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 <= (-3.901864712582398d-273)) then
        tmp = -a
    else
        tmp = (a - b) / sqrt(((a - b) / (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 <= -3.901864712582398e-273) {
		tmp = -a;
	} else {
		tmp = (a - b) / Math.sqrt(((a - b) / (a + b)));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if a <= -3.901864712582398e-273:
		tmp = -a
	else:
		tmp = (a - b) / math.sqrt(((a - b) / (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 <= -3.901864712582398e-273)
		tmp = Float64(-a);
	else
		tmp = Float64(Float64(a - b) / sqrt(Float64(Float64(a - b) / 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 <= -3.901864712582398e-273)
		tmp = -a;
	else
		tmp = (a - b) / sqrt(((a - b) / (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, -3.901864712582398e-273], (-a), N[(N[(a - b), $MachinePrecision] / N[Sqrt[N[(N[(a - b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	32.0
Target	0.5
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if a < -3.90186471258239784e-273

   1. Initial program 32.2

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

   2. Taylor expanded in a around -inf 0.7

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

   3. Simplified 0.7

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

      [Start] 0.7	\[ -1 \cdot a \]
      mul-1-neg [=>] 0.7	\[ \color{blue}{-a} \]

   if -3.90186471258239784e-273 < a

   1. Initial program 31.9

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

   2. Applied egg-rr 0.6

      \[\leadsto \color{blue}{\sqrt{a - b} \cdot \sqrt{a + b}} \]

   3. Applied egg-rr 21.5

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{b + a}}}} \]
      Proof

      [Start] 21.5	\[ \frac{\left(a - b\right) \cdot \sqrt{a + b}}{\sqrt{a - b}} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{a + b}}}} \]
      +-commutative [=>] 0.1	\[ \frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{\color{blue}{b + a}}}} \]

   5. Applied egg-rr 0.1

      \[\leadsto \frac{a - b}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{a - b}{a + b}}\right)} - 1}} \]

   6. Simplified 0.1

      \[\leadsto \frac{a - b}{\color{blue}{\sqrt{\frac{a - b}{a + b}}}} \]
      Proof

      [Start] 0.1	\[ \frac{a - b}{e^{\mathsf{log1p}\left(\sqrt{\frac{a - b}{a + b}}\right)} - 1} \]
      expm1-def [=>] 0.1	\[ \frac{a - b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{a - b}{a + b}}\right)\right)}} \]
      expm1-log1p [=>] 0.1	\[ \frac{a - b}{\color{blue}{\sqrt{\frac{a - b}{a + b}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.901864712582398 \cdot 10^{-273}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -6.203102982813514 \cdot 10^{-298}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -6.203102982813514 \cdot 10^{-298}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 3
Error	32.0
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023039 
(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))))