bug366, discussion (missed optimization)

Average Error: 49.87% → 0.95%

Time: 4.5s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -5e-236)
   (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 <= -5e-236) {
		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 <= -5e-236)
		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, -5e-236], 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	49.87%
Target	0.77%
Herbie	0.95%

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

Derivation

1. Split input into 2 regimes

2. if a < -4.9999999999999998e-236

   1. Initial program 48.92

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

   2. Taylor expanded in a around -inf 6.82

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

   3. Simplified 0.39

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

      [Start] 6.82	\[ 0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a \]
      fma-def [=>] 6.82	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{{b}^{2}}{a}, -1 \cdot a\right)} \]
      unpow2 [=>] 6.82	\[ \mathsf{fma}\left(0.5, \frac{\color{blue}{b \cdot b}}{a}, -1 \cdot a\right) \]
      associate-/l* [=>] 0.39	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{b}{\frac{a}{b}}}, -1 \cdot a\right) \]
      mul-1-neg [=>] 0.39	\[ \mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, \color{blue}{-a}\right) \]

   if -4.9999999999999998e-236 < a

   1. Initial program 50.81

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

   2. Applied egg-rr 2.33

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

   3. Applied egg-rr 34.99

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

   4. Simplified 1.6

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

      [Start] 34.99	\[ \frac{\left(a - b\right) \cdot \sqrt{a + b}}{\sqrt{a - b}} \]
      associate-/l* [=>] 1.6	\[ \color{blue}{\frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{a + b}}}} \]
      +-commutative [=>] 1.6	\[ \frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{\color{blue}{b + a}}}} \]

   5. Applied egg-rr 1.52

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

   6. Simplified 1.5

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

      [Start] 1.52	\[ \frac{a - b}{e^{\mathsf{log1p}\left(\sqrt{\frac{a - b}{a + b}}\right)} - 1} \]
      expm1-def [=>] 1.51	\[ \frac{a - b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{a - b}{a + b}}\right)\right)}} \]
      expm1-log1p [=>] 1.5	\[ \frac{a - b}{\color{blue}{\sqrt{\frac{a - b}{a + b}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.95

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - b}{\sqrt{\frac{a - b}{a + b}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.12%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-236}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 2
Error	1.36%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-236}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b \cdot -0.5}{\frac{a}{b}}\\ \end{array} \]

Alternative 3
Error	1.61%
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-236}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4
Error	50.44%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023121 
(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))))