bug366, discussion (missed optimization)

Percentage Accurate: 53.3% → 99.5%

Time: 4.1s

Alternatives: 3

Speedup: 1.0×

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: 53.3% 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, 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 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 48.3

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

5. Applied rewrites 48.3%

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

Alternative 2: 53.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
double code(double a_m, double b) {
	return sqrt((a_m * 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 = sqrt((a_m * a_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in b around 0

   \[\leadsto \sqrt{\color{blue}{{a}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 57.4

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

5. Applied rewrites 57.4%

   \[\leadsto \sqrt{\color{blue}{a \cdot a}} \]
6. Add Preprocessing

Alternative 3: 1.2% accurate, 8.0× 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 Float64(-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 57.0%

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

3. Taylor expanded in a around -inf

   \[\leadsto \color{blue}{-1 \cdot a} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(a\right)} \]

   2. lower-neg.f64 52.8

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

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{-a} \]
6. 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 2024270 
(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))))