bug366, discussion (missed optimization)

Percentage Accurate: 53.5% → 99.8%

Time: 7.3s

Alternatives: 4

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: 53.5% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (* (/ (- a_m b_m) (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 ((a_m - b_m) / 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 = ((a_m - b_m) / 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 ((a_m - b_m) / 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 ((a_m - b_m) / 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(Float64(Float64(a_m - b_m) / 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 = ((a_m - b_m) / 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[(N[(a$95$m - b$95$m), $MachinePrecision] / N[Sqrt[N[(a$95$m - b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(a$95$m + b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

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

   1. sqr-neg 61.7%

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

   2. difference-of-squares 62.1%

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

   3. sub-neg 62.1%

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

   4. distribute-rgt-out-- 61.7%

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

   5. cancel-sign-sub 61.7%

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

   6. distribute-rgt-in 62.1%

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

3. Simplified 62.1%

   \[\leadsto \color{blue}{\sqrt{\left(a - b\right) \cdot \left(a + b\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sqrt-prod 46.5%

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

6. Applied egg-rr 46.5%

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

   1. +-commutative 46.5%

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

8. Simplified 46.5%

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

   1. add0 46.5%

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

   2. flip-+ 46.5%

      \[\leadsto \color{blue}{\frac{\sqrt{a - b} \cdot \sqrt{a - b} - 0 \cdot 0}{\sqrt{a - b} - 0}} \cdot \sqrt{b + a} \]

   3. add-sqr-sqrt 46.8%

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

   4. metadata-eval 46.8%

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

   5. associate--r+ 46.8%

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

   6. add0 46.8%

      \[\leadsto \frac{a - \color{blue}{b}}{\sqrt{a - b} - 0} \cdot \sqrt{b + a} \]

   7. *-un-lft-identity 46.8%

      \[\leadsto \frac{a - b}{\color{blue}{1 \cdot \sqrt{a - b}} - 0} \cdot \sqrt{b + a} \]

   8. fma-neg 46.8%

      \[\leadsto \frac{a - b}{\color{blue}{\mathsf{fma}\left(1, \sqrt{a - b}, -0\right)}} \cdot \sqrt{b + a} \]

   9. metadata-eval 46.8%

      \[\leadsto \frac{a - b}{\mathsf{fma}\left(1, \sqrt{a - b}, \color{blue}{0}\right)} \cdot \sqrt{b + a} \]

   10. fma-define 46.8%

       \[\leadsto \frac{a - b}{\color{blue}{1 \cdot \sqrt{a - b} + 0}} \cdot \sqrt{b + a} \]

   11. *-un-lft-identity 46.8%

       \[\leadsto \frac{a - b}{\color{blue}{\sqrt{a - b}} + 0} \cdot \sqrt{b + a} \]

   12. add0 46.8%

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

10. Applied egg-rr 46.8%

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

11. Final simplification 46.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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)) / ((a_m - b_m) ** (-0.5d0))
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.pow((a_m - b_m), -0.5);
}

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.pow((a_m - b_m), -0.5)

Julia

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

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = sqrt((a_m + b_m)) / ((a_m - b_m) ^ -0.5);
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[Power[N[(a$95$m - b$95$m), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

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

   1. sqr-neg 61.7%

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

   2. difference-of-squares 62.1%

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

   3. sub-neg 62.1%

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

   4. distribute-rgt-out-- 61.7%

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

   5. cancel-sign-sub 61.7%

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

   6. distribute-rgt-in 62.1%

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

3. Simplified 62.1%

   \[\leadsto \color{blue}{\sqrt{\left(a - b\right) \cdot \left(a + b\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sqrt-prod 46.5%

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

6. Applied egg-rr 46.5%

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

   1. +-commutative 46.5%

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

8. Simplified 46.5%

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

   1. add0 46.5%

      \[\leadsto \color{blue}{\sqrt{a - b} \cdot \sqrt{b + a} + 0} \]

   2. fma-define 46.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{a - b}, \sqrt{b + a}, 0\right)} \]

   3. metadata-eval 46.5%

      \[\leadsto \mathsf{fma}\left(\sqrt{a - b}, \sqrt{b + a}, \color{blue}{-0}\right) \]

   4. fma-neg 46.5%

      \[\leadsto \color{blue}{\sqrt{a - b} \cdot \sqrt{b + a} - 0} \]

   5. add-cbrt-cube 35.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{a - b} \cdot \sqrt{a - b}\right) \cdot \sqrt{a - b}}} \cdot \sqrt{b + a} - 0 \]

   6. add-sqr-sqrt 35.5%

      \[\leadsto \sqrt[3]{\color{blue}{\left(a - b\right)} \cdot \sqrt{a - b}} \cdot \sqrt{b + a} - 0 \]

   7. cbrt-prod 46.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{a - b} \cdot \sqrt[3]{\sqrt{a - b}}\right)} \cdot \sqrt{b + a} - 0 \]

   8. associate-*l* 46.2%

      \[\leadsto \color{blue}{\sqrt[3]{a - b} \cdot \left(\sqrt[3]{\sqrt{a - b}} \cdot \sqrt{b + a}\right)} - 0 \]

   9. fma-neg 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{a - b}, \sqrt[3]{\sqrt{a - b}} \cdot \sqrt{b + a}, -0\right)} \]

   10. +-commutative 46.2%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{a - b}, \sqrt[3]{\sqrt{a - b}} \cdot \sqrt{\color{blue}{a + b}}, -0\right) \]

   11. metadata-eval 46.2%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{a - b}, \sqrt[3]{\sqrt{a - b}} \cdot \sqrt{a + b}, \color{blue}{0}\right) \]

10. Applied egg-rr 46.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{a - b}, \sqrt[3]{\sqrt{a - b}} \cdot \sqrt{a + b}, 0\right)} \]
11. Step-by-step derivation

    1. fma-undefine 46.2%

       \[\leadsto \color{blue}{\sqrt[3]{a - b} \cdot \left(\sqrt[3]{\sqrt{a - b}} \cdot \sqrt{a + b}\right) + 0} \]

    2. add0 46.2%

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

    3. *-commutative 46.2%

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

    4. associate-*r* 46.2%

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

    5. *-commutative 46.2%

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

    6. *-commutative 46.2%

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

    7. +-commutative 46.2%

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

12. Simplified 46.2%

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

    1. *-commutative 46.2%

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

    2. associate-*l* 46.2%

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

    3. cbrt-prod 35.5%

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

    4. add-sqr-sqrt 35.5%

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

    5. add-cbrt-cube 46.5%

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

    6. pow1/2 46.5%

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

    7. metadata-eval 46.5%

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

    8. pow-div 46.8%

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

    9. pow1 46.8%

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

    10. pow1/2 46.8%

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

    11. clear-num 46.6%

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

    12. un-div-inv 46.7%

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

    13. +-commutative 46.7%

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

    14. pow1/2 46.7%

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

    15. pow1 46.7%

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

    16. pow-div 46.6%

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

    17. metadata-eval 46.6%

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

14. Applied egg-rr 46.6%

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

    1. +-commutative 46.6%

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

16. Simplified 46.6%

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

17. Final simplification 46.6%

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

Alternative 3: 99.3% 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 61.7%

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

   1. sqr-neg 61.7%

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

   2. difference-of-squares 62.1%

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

   3. sub-neg 62.1%

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

   4. distribute-rgt-out-- 61.7%

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

   5. cancel-sign-sub 61.7%

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

   6. distribute-rgt-in 62.1%

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

3. Simplified 62.1%

   \[\leadsto \color{blue}{\sqrt{\left(a - b\right) \cdot \left(a + b\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sqrt-prod 46.5%

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

6. Applied egg-rr 46.5%

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

   1. +-commutative 46.5%

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

8. Simplified 46.5%

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

9. Final simplification 46.5%

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

Alternative 4: 98.9% 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 61.7%

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

   1. sqr-neg 61.7%

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

   2. difference-of-squares 62.1%

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

   3. sub-neg 62.1%

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

   4. distribute-rgt-out-- 61.7%

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

   5. cancel-sign-sub 61.7%

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

   6. distribute-rgt-in 62.1%

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

3. Simplified 62.1%

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

5. Taylor expanded in a around inf 46.3%

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

6. Final simplification 46.3%

   \[\leadsto a \]
7. Add Preprocessing

Developer target: 99.3% 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 2024034 
(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))))