bug366, discussion (missed optimization)

Average Error: 49.75% → 0.8%

Time: 4.5s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \sqrt{\frac{a - b}{a + b}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-241)
   (fma 0.5 (/ b (/ a b)) (- 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 <= -2e-241) {
		tmp = fma(0.5, (b / (a / b)), -a);
	} else {
		tmp = (a + b) * 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 <= -2e-241)
		tmp = fma(0.5, Float64(b / Float64(a / b)), Float64(-a));
	else
		tmp = Float64(Float64(a + b) * sqrt(Float64(Float64(a - b) / 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, -2e-241], N[(0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[Sqrt[N[(N[(a - b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	49.75%
Target	0.77%
Herbie	0.8%

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

Derivation

1. Split input into 2 regimes

2. if a < -1.9999999999999999e-241

   1. Initial program 48.84

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

   2. Taylor expanded in a around -inf 6.87

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

   3. Simplified 0.46

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

      [Start] 6.87	\[ 0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a \]
      fma-def [=>] 6.87	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{b}^{2}}{a}, -1 \cdot a\right)} \]
      unpow2 [=>] 6.87	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{b \cdot b}}{a}, -1 \cdot a\right) \]
      associate-/l* [=>] 0.46	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{b}{\frac{a}{b}}}, -1 \cdot a\right) \]
      mul-1-neg [=>] 0.46	\[ \mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, \color{blue}{-a}\right) \]

   if -1.9999999999999999e-241 < a

   1. Initial program 50.65

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

   2. Applied egg-rr 1.96

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

   3. Applied egg-rr 35.52

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

   4. Simplified 1.45

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

      [Start] 35.52	\[ \frac{\sqrt{a - b} \cdot \left(a + b\right)}{\sqrt{a + b}} \]
      associate-/l* [=>] 1.45	\[ \color{blue}{\frac{\sqrt{a - b}}{\frac{\sqrt{a + b}}{a + b}}} \]
      +-commutative [=>] 1.45	\[ \frac{\sqrt{a - b}}{\frac{\sqrt{\color{blue}{b + a}}}{a + b}} \]
      +-commutative [=>] 1.45	\[ \frac{\sqrt{a - b}}{\frac{\sqrt{b + a}}{\color{blue}{b + a}}} \]

   5. Applied egg-rr 1.15

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \sqrt{\frac{a - b}{a + b}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.06%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 2
Error	0.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-278}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 3
Error	1.6%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-241}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4
Error	50.46%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023090 
(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))))