bug366, discussion (missed optimization)

Average Accuracy: 50.4% → 99.3%

Time: 6.1s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -1e-252)
   (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 <= -1e-252) {
		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 <= -1e-252)
		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, -1e-252], 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	50.4%
Target	99.2%
Herbie	99.3%

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

Derivation

1. Split input into 2 regimes

2. if a < -9.99999999999999943e-253

   1. Initial program 50.9%

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

   2. Taylor expanded in a around -inf 93.4%

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

   3. Simplified 99.5%

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

      [Start] 93.4	\[ 0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a \]
      fma-def [=>] 93.4	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{b}^{2}}{a}, -1 \cdot a\right)} \]
      unpow2 [=>] 93.4	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{b \cdot b}}{a}, -1 \cdot a\right) \]
      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 -9.99999999999999943e-253 < a

   1. Initial program 49.9%

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

   2. Applied egg-rr 98.4%

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

      [Start] 49.9	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 49.9	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]
      sqrt-prod [=>] 98.4	\[ \color{blue}{\sqrt{a + b} \cdot \sqrt{a - b}} \]
      *-commutative [=>] 98.4	\[ \color{blue}{\sqrt{a - b} \cdot \sqrt{a + b}} \]

   3. Applied egg-rr 66.0%

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

      [Start] 98.4	\[ \sqrt{a - b} \cdot \sqrt{a + b} \]
      flip-+ [=>] 49.4	\[ \sqrt{a - b} \cdot \sqrt{\color{blue}{\frac{a \cdot a - b \cdot b}{a - b}}} \]
      sqrt-div [=>] 49.7	\[ \sqrt{a - b} \cdot \color{blue}{\frac{\sqrt{a \cdot a - b \cdot b}}{\sqrt{a - b}}} \]
      difference-of-squares [=>] 49.7	\[ \sqrt{a - b} \cdot \frac{\sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}}{\sqrt{a - b}} \]
      sqrt-unprod [<=] 98.4	\[ \sqrt{a - b} \cdot \frac{\color{blue}{\sqrt{a + b} \cdot \sqrt{a - b}}}{\sqrt{a - b}} \]
      *-commutative [<=] 98.4	\[ \sqrt{a - b} \cdot \frac{\color{blue}{\sqrt{a - b} \cdot \sqrt{a + b}}}{\sqrt{a - b}} \]
      associate-*r/ [=>] 65.6	\[ \color{blue}{\frac{\sqrt{a - b} \cdot \left(\sqrt{a - b} \cdot \sqrt{a + b}\right)}{\sqrt{a - b}}} \]

   4. Applied egg-rr 70.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\right)} - 1} \]
      Proof

      [Start] 66.0	\[ \frac{\left(a - b\right) \cdot \sqrt{a + b}}{\sqrt{a - b}} \]
      expm1-log1p-u [=>] 62.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(a - b\right) \cdot \sqrt{a + b}}{\sqrt{a - b}}\right)\right)} \]
      expm1-udef [=>] 43.3	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(a - b\right) \cdot \sqrt{a + b}}{\sqrt{a - b}}\right)} - 1} \]
      associate-/l* [=>] 70.5	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{a + b}}}}\right)} - 1 \]
      sqrt-undiv [=>] 70.6	\[ e^{\mathsf{log1p}\left(\frac{a - b}{\color{blue}{\sqrt{\frac{a - b}{a + b}}}}\right)} - 1 \]

   5. Simplified 99.1%

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

      [Start] 70.6	\[ e^{\mathsf{log1p}\left(\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\right)} - 1 \]
      expm1-def [=>] 92.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\right)\right)} \]
      expm1-log1p [=>] 99.1	\[ \color{blue}{\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}} \]
      +-commutative [=>] 99.1	\[ \frac{a - b}{\sqrt{\frac{a - b}{\color{blue}{b + a}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - b\right) \cdot \sqrt{\frac{a + b}{a - b}}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 3
Accuracy	98.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-252}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 4
Accuracy	98.6%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-252}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 5
Accuracy	49.8%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023137 
(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))))