bug366, discussion (missed optimization)

Percentage Accurate: 53.2% → 99.4%

Time: 4.5s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\left(b \cdot \frac{b}{a}\right) \cdot 0.5 - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -1e-285) (- (* (* b (/ b a)) 0.5) a) (fma -0.5 (/ 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 <= -1e-285) {
		tmp = ((b * (b / a)) * 0.5) - a;
	} else {
		tmp = fma(-0.5, (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 <= -1e-285)
		tmp = Float64(Float64(Float64(b * Float64(b / a)) * 0.5) - a);
	else
		tmp = fma(-0.5, Float64(b / Float64(a / b)), a);
	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, -1e-285], N[(N[(N[(b * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - a), $MachinePrecision], N[(-0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Target

Original	53.2%
Target	99.2%
Herbie	99.4%

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

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000007e-285

   1. Initial program 51.1%

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

   2. Simplified 51.7%

      \[\leadsto \color{blue}{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \]
      Step-by-step derivation

      [Start] 51.1%	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 51.7%	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666}\right)}^{3}} \]
      Step-by-step derivation

      [Start] 51.7%	\[ \sqrt{\left(a + b\right) \cdot \left(a - b\right)} \]
      add-cube-cbrt [=>] 50.7%	\[ \color{blue}{\left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]
      pow3 [=>] 50.7%	\[ \color{blue}{{\left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right)}^{3}} \]
      pow1/3 [=>] 47.6%	\[ {\color{blue}{\left({\left(\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]
      pow1/2 [=>] 47.6%	\[ {\left({\color{blue}{\left({\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{0.5}\right)}}^{0.3333333333333333}\right)}^{3} \]
      difference-of-squares [<=] 47.1%	\[ {\left({\left({\color{blue}{\left(a \cdot a - b \cdot b\right)}}^{0.5}\right)}^{0.3333333333333333}\right)}^{3} \]
      pow-pow [=>] 47.1%	\[ {\color{blue}{\left({\left(a \cdot a - b \cdot b\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}}^{3} \]
      fma-neg [=>] 47.6%	\[ {\left({\color{blue}{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}^{3} \]
      metadata-eval [=>] 47.6%	\[ {\left({\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \]

   4. Taylor expanded in a around -inf 93.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{b}^{2}}{a}\right) + \left(0.3333333333333333 \cdot \frac{{b}^{2}}{a} + -1 \cdot a\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(b \cdot \frac{b}{a}\right) \cdot 0.5 - a} \]
      Step-by-step derivation

      [Start] 93.8%	\[ 0.16666666666666666 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{b}^{2}}{a}\right) + \left(0.3333333333333333 \cdot \frac{{b}^{2}}{a} + -1 \cdot a\right) \]
      associate-+r+ [=>] 93.8%	\[ \color{blue}{\left(0.16666666666666666 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{b}^{2}}{a}\right) + 0.3333333333333333 \cdot \frac{{b}^{2}}{a}\right) + -1 \cdot a} \]
      neg-mul-1 [<=] 93.8%	\[ \left(0.16666666666666666 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{b}^{2}}{a}\right) + 0.3333333333333333 \cdot \frac{{b}^{2}}{a}\right) + \color{blue}{\left(-a\right)} \]
      unsub-neg [=>] 93.8%	\[ \color{blue}{\left(0.16666666666666666 \cdot \left({1}^{0.3333333333333333} \cdot \frac{{b}^{2}}{a}\right) + 0.3333333333333333 \cdot \frac{{b}^{2}}{a}\right) - a} \]
      pow-base-1 [=>] 93.8%	\[ \left(0.16666666666666666 \cdot \left(\color{blue}{1} \cdot \frac{{b}^{2}}{a}\right) + 0.3333333333333333 \cdot \frac{{b}^{2}}{a}\right) - a \]
      associate-*r* [=>] 93.8%	\[ \left(\color{blue}{\left(0.16666666666666666 \cdot 1\right) \cdot \frac{{b}^{2}}{a}} + 0.3333333333333333 \cdot \frac{{b}^{2}}{a}\right) - a \]
      metadata-eval [=>] 93.8%	\[ \left(\color{blue}{0.16666666666666666} \cdot \frac{{b}^{2}}{a} + 0.3333333333333333 \cdot \frac{{b}^{2}}{a}\right) - a \]
      unpow2 [=>] 93.8%	\[ \left(0.16666666666666666 \cdot \frac{\color{blue}{b \cdot b}}{a} + 0.3333333333333333 \cdot \frac{{b}^{2}}{a}\right) - a \]
      unpow2 [=>] 93.8%	\[ \left(0.16666666666666666 \cdot \frac{b \cdot b}{a} + 0.3333333333333333 \cdot \frac{\color{blue}{b \cdot b}}{a}\right) - a \]
      distribute-rgt-out [=>] 93.8%	\[ \color{blue}{\frac{b \cdot b}{a} \cdot \left(0.16666666666666666 + 0.3333333333333333\right)} - a \]
      associate-*r/ [<=] 100.0%	\[ \color{blue}{\left(b \cdot \frac{b}{a}\right)} \cdot \left(0.16666666666666666 + 0.3333333333333333\right) - a \]
      metadata-eval [=>] 100.0%	\[ \left(b \cdot \frac{b}{a}\right) \cdot \color{blue}{0.5} - a \]

   if -1.00000000000000007e-285 < a

   1. Initial program 46.9%

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

   2. Simplified 47.2%

      \[\leadsto \color{blue}{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \]
      Step-by-step derivation

      [Start] 46.9%	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 47.2%	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Taylor expanded in a around inf 95.7%

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

   4. Simplified 99.7%

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

      [Start] 95.7%	\[ 0.5 \cdot \left(b + -1 \cdot b\right) + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      fma-def [=>] 95.7%	\[ \color{blue}{\mathsf{fma}\left(0.5, b + -1 \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
      distribute-rgt1-in [=>] 95.7%	\[ \mathsf{fma}\left(0.5, \color{blue}{\left(-1 + 1\right) \cdot b}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      metadata-eval [=>] 95.7%	\[ \mathsf{fma}\left(0.5, \color{blue}{0} \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      mul0-lft [=>] 95.7%	\[ \mathsf{fma}\left(0.5, \color{blue}{0}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      fma-udef [=>] 95.7%	\[ \color{blue}{0.5 \cdot 0 + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
      metadata-eval [=>] 95.7%	\[ \color{blue}{0} + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      +-lft-identity [=>] 95.7%	\[ \color{blue}{0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\left(b \cdot \frac{b}{a}\right) \cdot 0.5 - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\left(b \cdot \frac{b}{a}\right) \cdot 0.5 - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	708

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

Alternative 3
Accuracy	98.8%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-285}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4
Accuracy	49.7%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023167 
(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))))