bug366, discussion (missed optimization)

Average Accuracy: 49.5% → 99.5%

Time: 4.8s

Precision: binary64

Cost: 7044

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -4e-304)
   (fma 0.5 (/ b (/ a b)) (- a))
   (+ a (/ (* b -0.5) (/ 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 <= -4e-304) {
		tmp = fma(0.5, (b / (a / b)), -a);
	} else {
		tmp = a + ((b * -0.5) / (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 <= -4e-304)
		tmp = fma(0.5, Float64(b / Float64(a / b)), Float64(-a));
	else
		tmp = Float64(a + Float64(Float64(b * -0.5) / 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, -4e-304], N[(0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision], N[(a + N[(N[(b * -0.5), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	49.5%
Target	99.2%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999988e-304

   1. Initial program 49.7%

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

   2. Taylor expanded in a around -inf 93.2%

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

   3. Simplified 99.5%

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

      [Start] 93.2	\[ 0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a \]
      fma-def [=>] 93.2	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{b}^{2}}{a}, -1 \cdot a\right)} \]
      unpow2 [=>] 93.2	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{b \cdot b}}{a}, -1 \cdot a\right) \]
      associate-/l* [=>] 99.5	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{b}{\frac{a}{b}}}, -1 \cdot a\right) \]
      mul-1-neg [=>] 99.5	\[ \mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, \color{blue}{-a}\right) \]

   if -3.99999999999999988e-304 < a

   1. Initial program 49.3%

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

   2. Taylor expanded in a around inf 93.5%

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

   3. Simplified 93.5%

      \[\leadsto \color{blue}{a + -0.5 \cdot \frac{b \cdot b}{a}} \]
      Proof

      [Start] 93.5	\[ a + -0.5 \cdot \frac{{b}^{2}}{a} \]
      unpow2 [=>] 93.5	\[ a + -0.5 \cdot \frac{\color{blue}{b \cdot b}}{a} \]

   4. Applied egg-rr 99.4%

      \[\leadsto a + \color{blue}{\frac{b \cdot -0.5}{\frac{a}{b}}} \]
      Proof

      [Start] 93.5	\[ a + -0.5 \cdot \frac{b \cdot b}{a} \]
      *-commutative [=>] 93.5	\[ a + \color{blue}{\frac{b \cdot b}{a} \cdot -0.5} \]
      associate-/l* [=>] 99.4	\[ a + \color{blue}{\frac{b}{\frac{a}{b}}} \cdot -0.5 \]
      associate-*l/ [=>] 99.4	\[ a + \color{blue}{\frac{b \cdot -0.5}{\frac{a}{b}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-288}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 2
Accuracy	98.9%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-288}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 3
Accuracy	49.9%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023138 
(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))))