bug366, discussion (missed optimization)

Percentage Accurate: 52.7% → 99.5%

Time: 6.6s

Alternatives: 3

Speedup: 24.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return sqrt(((a * a) - (b * b)));
}

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

Java

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

Python

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

Julia

function code(a, b)
	return sqrt(Float64(Float64(a * a) - Float64(b * b)))
end

MATLAB

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

Wolfram

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

Initial Program: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return sqrt(((a * a) - (b * b)));
}

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

Java

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

Python

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

Julia

function code(a, b)
	return sqrt(Float64(Float64(a * a) - Float64(b * b)))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \frac{\sqrt{b + a\_m}}{{\left(a\_m - b\right)}^{-0.5}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 (/ (sqrt (+ b a_m)) (pow (- a_m b) -0.5)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	return sqrt((b + a_m)) / pow((a_m - b), -0.5);
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = sqrt((b + a_m)) / ((a_m - b) ** (-0.5d0))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return Math.sqrt((b + a_m)) / Math.pow((a_m - b), -0.5);
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return math.sqrt((b + a_m)) / math.pow((a_m - b), -0.5)

Julia

a_m = abs(a)
function code(a_m, b)
	return Float64(sqrt(Float64(b + a_m)) / (Float64(a_m - b) ^ -0.5))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b)
	tmp = sqrt((b + a_m)) / ((a_m - b) ^ -0.5);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(N[Sqrt[N[(b + a$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[N[(a$95$m - b), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\sqrt{a \cdot a - b \cdot b} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. *-commutative N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower--.f64 N/A

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

   13. pow1/2 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 52.3

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

4. Applied rewrites 52.3%

   \[\leadsto \color{blue}{\sqrt{a - b} \cdot \sqrt{b + a}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

      \[\leadsto \sqrt{a - b} \cdot \sqrt{\color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}}} \]

   4. clear-num N/A

      \[\leadsto \sqrt{a - b} \cdot \sqrt{\color{blue}{\frac{1}{\frac{b \cdot b + \left(a \cdot a - b \cdot a\right)}{{b}^{3} + {a}^{3}}}}} \]

   5. sqrt-div N/A

      \[\leadsto \sqrt{a - b} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b \cdot b + \left(a \cdot a - b \cdot a\right)}{{b}^{3} + {a}^{3}}}}} \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{a - b} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{b \cdot b + \left(a \cdot a - b \cdot a\right)}{{b}^{3} + {a}^{3}}}} \]

   7. clear-num N/A

      \[\leadsto \sqrt{a - b} \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}}}}} \]

   8. flip3-+ N/A

      \[\leadsto \sqrt{a - b} \cdot \frac{1}{\sqrt{\frac{1}{\color{blue}{b + a}}}} \]

   9. lift-+.f64 N/A

      \[\leadsto \sqrt{a - b} \cdot \frac{1}{\sqrt{\frac{1}{\color{blue}{b + a}}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \sqrt{a - b} \cdot \color{blue}{\frac{1}{\sqrt{\frac{1}{b + a}}}} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \sqrt{a - b} \cdot \frac{1}{\color{blue}{\sqrt{\frac{1}{b + a}}}} \]

   12. inv-pow N/A

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

   13. lower-pow.f64 52.4

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

6. Applied rewrites 52.4%

   \[\leadsto \sqrt{a - b} \cdot \color{blue}{\frac{1}{\sqrt{{\left(b + a\right)}^{-1}}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 52.4

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

   7. lift-sqrt.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. sqrt-pow1 N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval 52.4

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

8. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{{\left(b + a\right)}^{-0.5}}{\sqrt{a - b}}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

   9. pow1/2 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{b + a}}}{\frac{1}{\sqrt{a - b}}} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt{b + a}}}{\frac{1}{\sqrt{a - b}}} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\sqrt{\color{blue}{b + a}}}{\frac{1}{\sqrt{a - b}}} \]

   12. +-commutative N/A

       \[\leadsto \frac{\sqrt{\color{blue}{a + b}}}{\frac{1}{\sqrt{a - b}}} \]

   13. lower-+.f64 N/A

       \[\leadsto \frac{\sqrt{\color{blue}{a + b}}}{\frac{1}{\sqrt{a - b}}} \]

   14. lift-sqrt.f64 N/A

       \[\leadsto \frac{\sqrt{a + b}}{\frac{1}{\color{blue}{\sqrt{a - b}}}} \]

   15. pow1/2 N/A

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

   16. pow-flip N/A

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

   17. metadata-eval N/A

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

   18. lower-pow.f64 52.5

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

10. Applied rewrites 52.5%

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

11. Final simplification 52.5%

    \[\leadsto \frac{\sqrt{b + a}}{{\left(a - b\right)}^{-0.5}} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \mathsf{fma}\left(\frac{-0.5}{a\_m} \cdot b, b, a\_m\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 (fma (* (/ -0.5 a_m) b) b a_m))

C

a_m = fabs(a);
double code(double a_m, double b) {
	return fma(((-0.5 / a_m) * b), b, a_m);
}

Julia

a_m = abs(a)
function code(a_m, b)
	return fma(Float64(Float64(-0.5 / a_m) * b), b, a_m)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(N[(N[(-0.5 / a$95$m), $MachinePrecision] * b), $MachinePrecision] * b + a$95$m), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\sqrt{a \cdot a - b \cdot b} \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{a + \frac{-1}{2} \cdot \frac{{b}^{2}}{a}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{a}\right)\right) \cdot \color{blue}{\left(b \cdot b\right)} + a \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{a}\right)\right) \cdot b\right) \cdot b} + a \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{a}\right)\right) \cdot b, b, a\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{a}\right)\right) \cdot b}, b, a\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{a}}\right)\right) \cdot b, b, a\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{a}\right)\right) \cdot b, b, a\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{a}} \cdot b, b, a\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2}}}{a} \cdot b, b, a\right) \]

   16. lower-/.f64 53.0

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

5. Applied rewrites 53.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{a} \cdot b, b, a\right)} \]
6. Add Preprocessing

Alternative 3: 99.0% accurate, 24.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a\_m \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 a_m)

C

a_m = fabs(a);
double code(double a_m, double b) {
	return a_m;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = a_m
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return a_m;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return a_m

Julia

a_m = abs(a)
function code(a_m, b)
	return a_m
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b)
	tmp = a_m;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := a$95$m

Derivation

1. Initial program 51.7%

   \[\sqrt{a \cdot a - b \cdot b} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 52.4

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

4. Applied rewrites 52.4%

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

5. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot \frac{b + -1 \cdot b}{a}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. distribute-rgt1-in N/A

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

   4. metadata-eval N/A

      \[\leadsto a \cdot \left(\frac{\frac{1}{2} \cdot \left(\color{blue}{0} \cdot b\right)}{a} + 1\right) \]

   5. mul0-lft N/A

      \[\leadsto a \cdot \left(\frac{\frac{1}{2} \cdot \color{blue}{0}}{a} + 1\right) \]

   6. metadata-eval N/A

      \[\leadsto a \cdot \left(\frac{\color{blue}{0}}{a} + 1\right) \]

   7. mul0-lft N/A

      \[\leadsto a \cdot \left(\frac{\color{blue}{0 \cdot b}}{a} + 1\right) \]

   8. associate-*r/ N/A

      \[\leadsto a \cdot \left(\color{blue}{0 \cdot \frac{b}{a}} + 1\right) \]

   9. mul0-lft N/A

      \[\leadsto a \cdot \left(\color{blue}{0} + 1\right) \]

   10. metadata-eval N/A

       \[\leadsto a \cdot \color{blue}{1} \]

   11. *-rgt-identity 52.8

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

7. Applied rewrites 52.8%

   \[\leadsto \color{blue}{a} \]
8. Add Preprocessing

Developer Target 1: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|a\right| + \left|b\right|} \cdot \sqrt{\left|a\right| - \left|b\right|} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return sqrt((fabs(a) + fabs(b))) * sqrt((fabs(a) - fabs(b)));
}

Fortran

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

Java

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

Python

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

Julia

function code(a, b)
	return Float64(sqrt(Float64(abs(a) + abs(b))) * sqrt(Float64(abs(a) - abs(b))))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024331 
(FPCore (a b)
  :name "bug366, discussion (missed optimization)"
  :precision binary64

  :alt
  (! :herbie-platform default (let* ((fa (fabs a)) (fb (fabs b))) (* (sqrt (+ fa fb)) (sqrt (- fa fb)))))

  (sqrt (- (* a a) (* b b))))