bug366, discussion (missed optimization)

Average Accuracy: 50.3% → 75.0%

Time: 2.2s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 0:\\ \;\;\;\;a\\ \mathbf{elif}\;a \cdot a \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 0.0) a (if (<= (* a a) 2e+304) (sqrt (* (+ a 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 * a) <= 0.0) {
		tmp = a;
	} else if ((a * a) <= 2e+304) {
		tmp = sqrt(((a + b) * (a - b)));
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(((a * a) - (b * b)))
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 0.0d0) then
        tmp = a
    else if ((a * a) <= 2d+304) then
        tmp = sqrt(((a + b) * (a - b)))
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	return Math.sqrt(((a * a) - (b * b)));
}

↓

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 0.0) {
		tmp = a;
	} else if ((a * a) <= 2e+304) {
		tmp = Math.sqrt(((a + b) * (a - b)));
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

↓

def code(a, b):
	tmp = 0
	if (a * a) <= 0.0:
		tmp = a
	elif (a * a) <= 2e+304:
		tmp = math.sqrt(((a + b) * (a - b)))
	else:
		tmp = 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 (Float64(a * a) <= 0.0)
		tmp = a;
	elseif (Float64(a * a) <= 2e+304)
		tmp = sqrt(Float64(Float64(a + b) * Float64(a - b)));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(((a * a) - (b * b)));
end

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 0.0)
		tmp = a;
	elseif ((a * a) <= 2e+304)
		tmp = sqrt(((a + b) * (a - b)));
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := N[Sqrt[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 0.0], a, If[LessEqual[N[(a * a), $MachinePrecision], 2e+304], N[Sqrt[N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], a]]

Target

Original	50.3%
Target	99.2%
Herbie	75.0%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 0.0 or 1.9999999999999999e304 < (*.f64 a a)

   1. Initial program 1.7%

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

   2. Simplified 1.7%

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

      [Start] 1.7	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 1.7	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Taylor expanded in a around inf 50.8%

      \[\leadsto \color{blue}{a} \]

   if 0.0 < (*.f64 a a) < 1.9999999999999999e304

   1. Initial program 99.5%

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

   2. Simplified 99.5%

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

      [Start] 99.5	\[ \sqrt{a \cdot a - b \cdot b} \]
      difference-of-squares [=>] 99.5	\[ \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 0:\\ \;\;\;\;a\\ \mathbf{elif}\;a \cdot a \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	49.6%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023151 
(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))))