bug366, discussion (missed optimization)

Percentage Accurate: 53.2% → 99.2%

Time: 4.1s

Alternatives: 3

Speedup: 107.0×

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: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \sqrt{a\_m \cdot \left(1 + \frac{b\_m}{a\_m}\right)} \cdot \sqrt{a\_m - b\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (* (sqrt (* a_m (+ 1.0 (/ b_m a_m)))) (sqrt (- a_m b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	return sqrt((a_m * (1.0 + (b_m / a_m)))) * sqrt((a_m - b_m));
}

Fortran

a_m = abs(a)
b_m = abs(b)
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = sqrt((a_m * (1.0d0 + (b_m / a_m)))) * sqrt((a_m - b_m))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m) {
	return Math.sqrt((a_m * (1.0 + (b_m / a_m)))) * Math.sqrt((a_m - b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	return math.sqrt((a_m * (1.0 + (b_m / a_m)))) * math.sqrt((a_m - b_m))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	return Float64(sqrt(Float64(a_m * Float64(1.0 + Float64(b_m / a_m)))) * sqrt(Float64(a_m - b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = sqrt((a_m * (1.0 + (b_m / a_m)))) * sqrt((a_m - b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := N[(N[Sqrt[N[(a$95$m * N[(1.0 + N[(b$95$m / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a$95$m - b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. pow1/2 53.4%

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

   2. difference-of-squares 53.9%

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

   3. unpow-prod-down 51.2%

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

4. Applied egg-rr 51.2%

   \[\leadsto \color{blue}{{\left(a + b\right)}^{0.5} \cdot {\left(a - b\right)}^{0.5}} \]
5. Step-by-step derivation

   1. unpow1/2 51.2%

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

   2. unpow1/2 51.2%

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

6. Simplified 51.2%

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

7. Taylor expanded in a around inf 51.2%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

a_m = abs(a)
b_m = abs(b)
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((a_m + b_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. pow1/2 53.4%

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

   2. difference-of-squares 53.9%

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

   3. unpow-prod-down 51.2%

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

4. Applied egg-rr 51.2%

   \[\leadsto \color{blue}{{\left(a + b\right)}^{0.5} \cdot {\left(a - b\right)}^{0.5}} \]
5. Step-by-step derivation

   1. unpow1/2 51.2%

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

   2. unpow1/2 51.2%

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

6. Simplified 51.2%

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

7. Final simplification 51.2%

   \[\leadsto \sqrt{a - b} \cdot \sqrt{a + b} \]
8. Add Preprocessing

Alternative 3: 99.2% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

a_m = abs(a)
b_m = abs(b)
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in a around inf 52.2%

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

Developer Target 1: 99.2% accurate, 0.2× 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 2024157 
(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))))