bug366, discussion (missed optimization)

Average Error: 31.4 → 0.5

Time: 4.3s

Precision: binary64

Cost: 13380

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1.9646778310978972 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -1.9646778310978972e-296)
   (fma 0.5 (/ b (/ a b)) (- 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 <= -1.9646778310978972e-296) {
		tmp = fma(0.5, (b / (a / b)), -a);
	} else {
		tmp = sqrt((a - b)) * 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 <= -1.9646778310978972e-296)
		tmp = fma(0.5, Float64(b / Float64(a / b)), Float64(-a));
	else
		tmp = Float64(sqrt(Float64(a - b)) * sqrt(Float64(a + b)));
	end
	return 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, -1.9646778310978972e-296], N[(0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision], N[(N[Sqrt[N[(a - b), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	31.4
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 < -1.96467783109789725e-296

   1. Initial program 31.4

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

   2. Taylor expanded in a around -inf 4.5

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

   3. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)} \]
      Proof

      [Start] 4.5	\[ 0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a \]
      fma-def [=>] 4.5	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{b}^{2}}{a}, -1 \cdot a\right)} \]
      unpow2 [=>] 4.5	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{b \cdot b}}{a}, -1 \cdot a\right) \]
      associate-/l* [=>] 0.4	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{b}{\frac{a}{b}}}, -1 \cdot a\right) \]
      mul-1-neg [=>] 0.4	\[ \mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, \color{blue}{-a}\right) \]

   if -1.96467783109789725e-296 < a

   1. Initial program 31.4

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

   2. Applied egg-rr 0.5

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9646778310978972 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	7044

\[\begin{array}{l} \mathbf{if}\;a \leq -1.9646778310978972 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -1.9646778310978972 \cdot 10^{-296}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -1.9646778310978972 \cdot 10^{-296}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4
Error	31.8
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023057 
(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))))