bug366, discussion (missed optimization)

Percentage Accurate: 53.2% → 100.0%

Time: 5.9s

Alternatives: 4

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: 53.2% 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: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\sqrt{a \cdot a - b \cdot b} \]
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 45.8

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

4. Applied rewrites 45.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. difference-of-squares N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. associate-/l* N/A

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

   11. sqrt-prod N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-/.f64 45.8

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

6. Applied rewrites 45.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 46.7

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

8. Applied rewrites 46.7%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m)
 :precision binary64
 (* (sqrt (- a_m b_m)) (sqrt (+ b_m a_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\sqrt{a \cdot a - b \cdot b} \]
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 45.8

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

4. Applied rewrites 45.8%

   \[\leadsto \color{blue}{\sqrt{a - b} \cdot \sqrt{b + a}} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\sqrt{a \cdot a - b \cdot b} \]
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 46.3

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

5. Applied rewrites 46.3%

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

Alternative 4: 98.9% accurate, 24.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\sqrt{a \cdot a - b \cdot b} \]
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 45.8

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

4. Applied rewrites 45.8%

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

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 45.9

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

7. Applied rewrites 45.9%

   \[\leadsto \color{blue}{a} \]
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 2024322 
(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))))