bug366, discussion (missed optimization)

Average Accuracy: 53.3% → 99.5%

Time: 5.0s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-289}:\\ \;\;\;\;\frac{b \cdot 0.5}{\frac{a}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -5e-289) (- (/ (* b 0.5) (/ a b)) 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 <= -5e-289) {
		tmp = ((b * 0.5) / (a / b)) - 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 <= -5e-289)
		tmp = Float64(Float64(Float64(b * 0.5) / Float64(a / b)) - 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, -5e-289], N[(N[(N[(b * 0.5), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], N[(-0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Target

Original	53.3%
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 < -5.00000000000000029e-289

   1. Initial program 51.3%

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

   2. Simplified 51.8%

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

      [Start] 51.3	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 51.8	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Applied egg-rr 51.5%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.25}\right)}^{2}} \]
      Proof

      [Start] 51.8	\[ \sqrt{\left(a + b\right) \cdot \left(a - b\right)} \]
      add-sqr-sqrt [=>] 51.5	\[ \color{blue}{\sqrt{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]
      pow2 [=>] 51.5	\[ \color{blue}{{\left(\sqrt{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right)}^{2}} \]
      pow1/2 [=>] 51.5	\[ {\left(\sqrt{\color{blue}{{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{0.5}}}\right)}^{2} \]
      sqrt-pow1 [=>] 51.5	\[ {\color{blue}{\left({\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]
      difference-of-squares [<=] 51.0	\[ {\left({\color{blue}{\left(a \cdot a - b \cdot b\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]
      fma-neg [=>] 51.5	\[ {\left({\color{blue}{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]
      metadata-eval [=>] 51.5	\[ {\left({\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Taylor expanded in a around -inf 94.2%

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

   5. Simplified 94.2%

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

      [Start] 94.2	\[ 0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a \]
      mul-1-neg [=>] 94.2	\[ 0.5 \cdot \frac{{b}^{2}}{a} + \color{blue}{\left(-a\right)} \]
      unsub-neg [=>] 94.2	\[ \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} - a} \]
      *-commutative [=>] 94.2	\[ \color{blue}{\frac{{b}^{2}}{a} \cdot 0.5} - a \]
      unpow2 [=>] 94.2	\[ \frac{\color{blue}{b \cdot b}}{a} \cdot 0.5 - a \]

   6. Applied egg-rr 99.7%

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

      [Start] 94.2	\[ \frac{b \cdot b}{a} \cdot 0.5 - a \]
      associate-/l* [=>] 99.7	\[ \color{blue}{\frac{b}{\frac{a}{b}}} \cdot 0.5 - a \]
      associate-*l/ [=>] 99.7	\[ \color{blue}{\frac{b \cdot 0.5}{\frac{a}{b}}} - a \]

   if -5.00000000000000029e-289 < a

   1. Initial program 54.5%

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

   2. Simplified 55.2%

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

      [Start] 54.5	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 55.2	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Taylor expanded in a around inf 92.8%

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

   4. Simplified 100.0%

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

      [Start] 92.8	\[ 0.5 \cdot \left(b + -1 \cdot b\right) + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      fma-def [=>] 92.8	\[ \color{blue}{\mathsf{fma}\left(0.5, b + -1 \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
      distribute-rgt1-in [=>] 92.8	\[ \mathsf{fma}\left(0.5, \color{blue}{\left(-1 + 1\right) \cdot b}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      metadata-eval [=>] 92.8	\[ \mathsf{fma}\left(0.5, \color{blue}{0} \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      mul0-lft [=>] 92.8	\[ \mathsf{fma}\left(0.5, \color{blue}{0}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      fma-udef [=>] 92.8	\[ \color{blue}{0.5 \cdot 0 + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
      metadata-eval [=>] 92.8	\[ \color{blue}{0} + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      +-lft-identity [=>] 92.8	\[ \color{blue}{0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-289}:\\ \;\;\;\;\frac{b \cdot 0.5}{\frac{a}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-289}:\\ \;\;\;\;\frac{b \cdot 0.5}{\frac{a}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-289}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 3
Accuracy	50.5%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023160 
(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))))