bug366, discussion (missed optimization)

Percentage Accurate: 54.0% → 76.9%

Time: 7.4s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -5e-287) (sqrt (* (+ a b) (- a b))) (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-287) {
		tmp = sqrt(((a + b) * (a - b)));
	} 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-287)
		tmp = sqrt(Float64(Float64(a + b) * Float64(a - b)));
	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-287], N[Sqrt[N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Target

Original	54.0%
Target	99.2%
Herbie	76.9%

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

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000025e-287

   1. Initial program 53.7%

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

   2. Simplified 54.3%

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

      [Start] 53.7%	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 54.3%	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   if -5.00000000000000025e-287 < a

   1. Initial program 57.1%

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

   2. Simplified 57.6%

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

      [Start] 57.1%	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 57.6%	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Taylor expanded in a around inf 93.0%

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

   4. Simplified 99.9%

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

      [Start] 93.0%	\[ 0.5 \cdot \left(b + -1 \cdot b\right) + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      fma-def [=>] 93.0%	\[ \color{blue}{\mathsf{fma}\left(0.5, b + -1 \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
      distribute-rgt1-in [=>] 93.0%	\[ \mathsf{fma}\left(0.5, \color{blue}{\left(-1 + 1\right) \cdot b}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      metadata-eval [=>] 93.0%	\[ \mathsf{fma}\left(0.5, \color{blue}{0} \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      mul0-lft [=>] 93.0%	\[ \mathsf{fma}\left(0.5, \color{blue}{0}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      fma-udef [=>] 93.0%	\[ \color{blue}{0.5 \cdot 0 + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
      metadata-eval [=>] 93.0%	\[ \color{blue}{0} + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]
      +-lft-identity [=>] 93.0%	\[ \color{blue}{0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a} \]

3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	76.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

Alternative 2
Accuracy	76.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 3
Accuracy	50.6%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023272 
(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))))