bug366, discussion (missed optimization)

Percentage Accurate: 53.8% → 99.3%

Time: 5.5s

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: 53.8% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \sqrt{b_m + a_m} \cdot \left({\left(a_m - b_m\right)}^{0.375} \cdot {\left(a_m - b_m\right)}^{0.125}\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (* (sqrt (+ b_m a_m)) (* (pow (- a_m b_m) 0.375) (pow (- a_m b_m) 0.125))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	return sqrt((b_m + a_m)) * (pow((a_m - b_m), 0.375) * pow((a_m - b_m), 0.125));
}

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((b_m + a_m)) * (((a_m - b_m) ** 0.375d0) * ((a_m - b_m) ** 0.125d0))
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((b_m + a_m)) * (Math.pow((a_m - b_m), 0.375) * Math.pow((a_m - b_m), 0.125));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	return math.sqrt((b_m + a_m)) * (math.pow((a_m - b_m), 0.375) * math.pow((a_m - b_m), 0.125))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	return Float64(sqrt(Float64(b_m + a_m)) * Float64((Float64(a_m - b_m) ^ 0.375) * (Float64(a_m - b_m) ^ 0.125)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = sqrt((b_m + a_m)) * (((a_m - b_m) ^ 0.375) * ((a_m - b_m) ^ 0.125));
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[(b$95$m + a$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(a$95$m - b$95$m), $MachinePrecision], 0.375], $MachinePrecision] * N[Power[N[(a$95$m - b$95$m), $MachinePrecision], 0.125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

   1. pow1/2 50.6%

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

   2. difference-of-squares 51.2%

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

   3. unpow-prod-down 45.7%

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

3. Applied egg-rr 45.7%

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

   1. unpow1/2 45.7%

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

   2. unpow1/2 45.7%

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

   3. +-commutative 45.7%

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

5. Simplified 45.7%

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

   1. pow1/2 45.7%

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

   2. add-cube-cbrt 45.4%

      \[\leadsto \sqrt{b + a} \cdot {\color{blue}{\left(\left(\sqrt[3]{a - b} \cdot \sqrt[3]{a - b}\right) \cdot \sqrt[3]{a - b}\right)}}^{0.5} \]

   3. unpow-prod-down 45.4%

      \[\leadsto \sqrt{b + a} \cdot \color{blue}{\left({\left(\sqrt[3]{a - b} \cdot \sqrt[3]{a - b}\right)}^{0.5} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right)} \]

   4. pow2 45.4%

      \[\leadsto \sqrt{b + a} \cdot \left({\color{blue}{\left({\left(\sqrt[3]{a - b}\right)}^{2}\right)}}^{0.5} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

7. Applied egg-rr 45.4%

   \[\leadsto \sqrt{b + a} \cdot \color{blue}{\left({\left({\left(\sqrt[3]{a - b}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right)} \]
8. Step-by-step derivation

   1. unpow1/2 45.4%

      \[\leadsto \sqrt{b + a} \cdot \left(\color{blue}{\sqrt{{\left(\sqrt[3]{a - b}\right)}^{2}}} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   2. unpow2 45.4%

      \[\leadsto \sqrt{b + a} \cdot \left(\sqrt{\color{blue}{\sqrt[3]{a - b} \cdot \sqrt[3]{a - b}}} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   3. rem-sqrt-square 45.4%

      \[\leadsto \sqrt{b + a} \cdot \left(\color{blue}{\left|\sqrt[3]{a - b}\right|} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   4. rem-square-sqrt 45.3%

      \[\leadsto \sqrt{b + a} \cdot \left(\left|\color{blue}{\sqrt{\sqrt[3]{a - b}} \cdot \sqrt{\sqrt[3]{a - b}}}\right| \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   5. unpow1/2 45.3%

      \[\leadsto \sqrt{b + a} \cdot \left(\left|\color{blue}{{\left(\sqrt[3]{a - b}\right)}^{0.5}} \cdot \sqrt{\sqrt[3]{a - b}}\right| \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   6. unpow1/2 45.3%

      \[\leadsto \sqrt{b + a} \cdot \left(\left|{\left(\sqrt[3]{a - b}\right)}^{0.5} \cdot \color{blue}{{\left(\sqrt[3]{a - b}\right)}^{0.5}}\right| \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   7. fabs-sqr 45.3%

      \[\leadsto \sqrt{b + a} \cdot \left(\color{blue}{\left({\left(\sqrt[3]{a - b}\right)}^{0.5} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right)} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   8. unpow1/2 45.3%

      \[\leadsto \sqrt{b + a} \cdot \left(\left(\color{blue}{\sqrt{\sqrt[3]{a - b}}} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   9. unpow1/2 45.3%

      \[\leadsto \sqrt{b + a} \cdot \left(\left(\sqrt{\sqrt[3]{a - b}} \cdot \color{blue}{\sqrt{\sqrt[3]{a - b}}}\right) \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   10. rem-square-sqrt 45.4%

       \[\leadsto \sqrt{b + a} \cdot \left(\color{blue}{\sqrt[3]{a - b}} \cdot {\left(\sqrt[3]{a - b}\right)}^{0.5}\right) \]

   11. unpow1/2 45.4%

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

9. Simplified 45.4%

   \[\leadsto \sqrt{b + a} \cdot \color{blue}{\left(\sqrt[3]{a - b} \cdot \sqrt{\sqrt[3]{a - b}}\right)} \]
10. Step-by-step derivation

    1. pow1 45.4%

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

    2. pow1/2 45.4%

       \[\leadsto \sqrt{b + a} \cdot \left({\left(\sqrt[3]{a - b}\right)}^{1} \cdot \color{blue}{{\left(\sqrt[3]{a - b}\right)}^{0.5}}\right) \]

    3. metadata-eval 45.4%

       \[\leadsto \sqrt{b + a} \cdot \left({\left(\sqrt[3]{a - b}\right)}^{1} \cdot {\left(\sqrt[3]{a - b}\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}}\right) \]

    4. pow-prod-up 45.4%

       \[\leadsto \sqrt{b + a} \cdot \color{blue}{{\left(\sqrt[3]{a - b}\right)}^{\left(1 + 0.25 \cdot 2\right)}} \]

    5. metadata-eval 45.4%

       \[\leadsto \sqrt{b + a} \cdot {\left(\sqrt[3]{a - b}\right)}^{\left(1 + \color{blue}{0.5}\right)} \]

    6. metadata-eval 45.4%

       \[\leadsto \sqrt{b + a} \cdot {\left(\sqrt[3]{a - b}\right)}^{\color{blue}{1.5}} \]

11. Applied egg-rr 45.4%

    \[\leadsto \sqrt{b + a} \cdot \color{blue}{{\left(\sqrt[3]{a - b}\right)}^{1.5}} \]
12. Step-by-step derivation

    1. pow1/3 42.1%

       \[\leadsto \sqrt{b + a} \cdot {\color{blue}{\left({\left(a - b\right)}^{0.3333333333333333}\right)}}^{1.5} \]

    2. pow-pow 45.7%

       \[\leadsto \sqrt{b + a} \cdot \color{blue}{{\left(a - b\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]

    3. metadata-eval 45.7%

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

    4. metadata-eval 45.7%

       \[\leadsto \sqrt{b + a} \cdot {\left(a - b\right)}^{\color{blue}{\left(0.125 + 0.375\right)}} \]

    5. metadata-eval 45.7%

       \[\leadsto \sqrt{b + a} \cdot {\left(a - b\right)}^{\left(0.125 + \color{blue}{\left(0.125 + 0.25\right)}\right)} \]

    6. pow-prod-up 45.7%

       \[\leadsto \sqrt{b + a} \cdot \color{blue}{\left({\left(a - b\right)}^{0.125} \cdot {\left(a - b\right)}^{\left(0.125 + 0.25\right)}\right)} \]

    7. metadata-eval 45.7%

       \[\leadsto \sqrt{b + a} \cdot \left({\left(a - b\right)}^{0.125} \cdot {\left(a - b\right)}^{\color{blue}{0.375}}\right) \]

13. Applied egg-rr 45.7%

    \[\leadsto \sqrt{b + a} \cdot \color{blue}{\left({\left(a - b\right)}^{0.125} \cdot {\left(a - b\right)}^{0.375}\right)} \]
14. Step-by-step derivation

    1. *-commutative 45.7%

       \[\leadsto \sqrt{b + a} \cdot \color{blue}{\left({\left(a - b\right)}^{0.375} \cdot {\left(a - b\right)}^{0.125}\right)} \]

15. Simplified 45.7%

    \[\leadsto \sqrt{b + a} \cdot \color{blue}{\left({\left(a - b\right)}^{0.375} \cdot {\left(a - b\right)}^{0.125}\right)} \]

16. Final simplification 45.7%

    \[\leadsto \sqrt{b + a} \cdot \left({\left(a - b\right)}^{0.375} \cdot {\left(a - b\right)}^{0.125}\right) \]

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (* (sqrt (+ 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((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((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((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((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(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((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[(b$95$m + a$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a$95$m - b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

   1. pow1/2 50.6%

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

   2. difference-of-squares 51.2%

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

   3. unpow-prod-down 45.7%

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

3. Applied egg-rr 45.7%

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

   1. unpow1/2 45.7%

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

   2. unpow1/2 45.7%

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

   3. +-commutative 45.7%

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

5. Simplified 45.7%

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

6. Final simplification 45.7%

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

Alternative 3: 99.0% 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 50.6%

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

2. Taylor expanded in a around inf 45.6%

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

3. Final simplification 45.6%

   \[\leadsto a \]

Developer target: 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 2023315 
(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))))