bug366, discussion (missed optimization)

Percentage Accurate: 52.9% → 99.8%

Time: 6.1s

Alternatives: 5

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: 52.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. pow1/2 56.9%

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

   2. difference-of-squares 57.3%

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

   3. unpow-prod-down 53.9%

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

4. Applied egg-rr 53.9%

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

   1. unpow1/2 53.9%

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

   2. unpow1/2 53.9%

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

   3. +-commutative 53.9%

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

6. Simplified 53.9%

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

   1. add-sqr-sqrt 53.8%

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

   2. pow2 53.8%

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

   3. pow1/2 53.8%

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

   4. sqrt-pow1 53.8%

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

   5. metadata-eval 53.8%

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

8. Applied egg-rr 53.8%

   \[\leadsto \color{blue}{{\left({\left(b + a\right)}^{0.25}\right)}^{2}} \cdot \sqrt{a - b} \]
9. Step-by-step derivation

   1. pow-pow 53.9%

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

   2. metadata-eval 53.9%

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

   3. pow1/2 53.9%

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

   4. flip-+ 31.5%

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

   5. sqrt-div 0.0%

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

   6. difference-of-squares 0.0%

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

   7. sub-neg 0.0%

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

   8. mul-1-neg 0.0%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 0.0%

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

   11. mul-1-neg 0.0%

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

   12. mul-1-neg 0.0%

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

   13. sqr-neg 0.0%

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

   14. sqrt-prod 0.0%

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

   15. add-sqr-sqrt 0.0%

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

   16. sqrt-unprod 0.0%

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

   17. add-sqr-sqrt 0.0%

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

   18. +-commutative 0.0%

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

   19. sub-neg 0.0%

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

   20. mul-1-neg 0.0%

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

   21. add-sqr-sqrt 0.0%

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

   22. sqrt-unprod 30.8%

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

   23. mul-1-neg 30.8%

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

   24. mul-1-neg 30.8%

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

   25. sqr-neg 30.8%

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

10. Applied egg-rr 54.3%

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

    1. +-commutative 54.3%

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

    2. +-commutative 54.3%

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

12. Simplified 54.3%

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

13. Final simplification 54.3%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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 - b)) / ((b + a_m) ** (-0.5d0))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return Math.sqrt((a_m - b)) / Math.pow((b + a_m), -0.5);
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return math.sqrt((a_m - b)) / math.pow((b + a_m), -0.5)

Julia

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

MATLAB

a_m = abs(a);
function tmp = code(a_m, b)
	tmp = sqrt((a_m - b)) / ((b + a_m) ^ -0.5);
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. pow1/2 56.9%

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

   2. difference-of-squares 57.3%

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

   3. unpow-prod-down 53.9%

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

4. Applied egg-rr 53.9%

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

   1. unpow1/2 53.9%

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

   2. unpow1/2 53.9%

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

   3. +-commutative 53.9%

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

6. Simplified 53.9%

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

   1. add-sqr-sqrt 53.8%

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

   2. pow2 53.8%

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

   3. pow1/2 53.8%

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

   4. sqrt-pow1 53.8%

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

   5. metadata-eval 53.8%

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

8. Applied egg-rr 53.8%

   \[\leadsto \color{blue}{{\left({\left(b + a\right)}^{0.25}\right)}^{2}} \cdot \sqrt{a - b} \]
9. Step-by-step derivation

   1. pow-pow 53.9%

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

   2. metadata-eval 53.9%

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

   3. pow1/2 53.9%

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

   4. flip-+ 31.5%

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

   5. sqrt-div 0.0%

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

   6. difference-of-squares 0.0%

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

   7. sub-neg 0.0%

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

   8. mul-1-neg 0.0%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 0.0%

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

   11. mul-1-neg 0.0%

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

   12. mul-1-neg 0.0%

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

   13. sqr-neg 0.0%

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

   14. sqrt-prod 0.0%

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

   15. add-sqr-sqrt 0.0%

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

   16. sqrt-unprod 0.0%

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

   17. add-sqr-sqrt 0.0%

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

   18. +-commutative 0.0%

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

   19. sub-neg 0.0%

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

   20. mul-1-neg 0.0%

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

   21. add-sqr-sqrt 0.0%

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

   22. sqrt-unprod 30.8%

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

   23. mul-1-neg 30.8%

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

   24. mul-1-neg 30.8%

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

   25. sqr-neg 30.8%

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

10. Applied egg-rr 54.3%

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

    1. +-commutative 54.3%

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

    2. +-commutative 54.3%

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

12. Simplified 54.3%

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

    1. *-commutative 54.3%

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

    2. clear-num 54.1%

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

    3. un-div-inv 54.2%

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

    4. pow1/2 54.2%

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

    5. pow1 54.2%

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

    6. pow-div 54.1%

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

    7. metadata-eval 54.1%

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

14. Applied egg-rr 54.1%

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

15. Final simplification 54.1%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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((b + a_m)) * sqrt((a_m - b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. pow1/2 56.9%

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

   2. difference-of-squares 57.3%

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

   3. unpow-prod-down 53.9%

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

4. Applied egg-rr 53.9%

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

   1. unpow1/2 53.9%

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

   2. unpow1/2 53.9%

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

   3. +-commutative 53.9%

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

6. Simplified 53.9%

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

7. Final simplification 53.9%

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

Alternative 4: 99.6% accurate, 11.9× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
double code(double a_m, double b) {
	return a_m + ((b * -0.5) * (b / 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 + ((b * (-0.5d0)) * (b / a_m))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return a_m + ((b * -0.5) * (b / a_m));
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return a_m + ((b * -0.5) * (b / a_m))

Julia

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

MATLAB

a_m = abs(a);
function tmp = code(a_m, b)
	tmp = a_m + ((b * -0.5) * (b / a_m));
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. pow1/2 56.9%

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

   2. difference-of-squares 57.3%

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

   3. unpow-prod-down 53.9%

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

4. Applied egg-rr 53.9%

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

   1. unpow1/2 53.9%

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

   2. unpow1/2 53.9%

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

   3. +-commutative 53.9%

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

6. Simplified 53.9%

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

   1. add-cbrt-cube 38.8%

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

   2. add-sqr-sqrt 38.9%

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

   3. cbrt-prod 53.6%

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

8. Applied egg-rr 53.6%

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

9. Taylor expanded in b around 0 49.8%

   \[\leadsto \color{blue}{a + b \cdot \left(-0.5 \cdot \frac{b \cdot \left(1 + 0.25 \cdot \frac{{\left(a + -1 \cdot a\right)}^{2}}{{a}^{2}}\right)}{a} + 0.5 \cdot \frac{a + -1 \cdot a}{a}\right)} \]

11. Simplified 54.3%

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

12. Final simplification 54.3%

    \[\leadsto a + \left(b \cdot -0.5\right) \cdot \frac{b}{a} \]
13. Add Preprocessing

Alternative 5: 99.1% accurate, 107.0× 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 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 56.9%

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

3. Taylor expanded in a around inf 53.8%

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

4. Final simplification 53.8%

   \[\leadsto a \]
5. Add Preprocessing

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 2024078 
(FPCore (a b)
  :name "bug366, discussion (missed optimization)"
  :precision binary64

  :alt
  (* (sqrt (+ (fabs a) (fabs b))) (sqrt (- (fabs a) (fabs b))))

  (sqrt (- (* a a) (* b b))))