sqrt E (should all be same)

Percentage Accurate: 54.1% → 100.0%

Time: 3.2s

Alternatives: 4

Speedup: 3.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{{x}^{2} + {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))

C

double code(double x) {
	return sqrt((pow(x, 2.0) + pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((x ** 2.0d0) + (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, 2.0) + Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((math.pow(x, 2.0) + math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64((x ^ 2.0) + (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ 2.0) + (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{{x}^{2} + {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))

C

double code(double x) {
	return sqrt((pow(x, 2.0) + pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((x ** 2.0d0) + (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, 2.0) + Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((math.pow(x, 2.0) + math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64((x ^ 2.0) + (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ 2.0) + (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (hypot x x))

C

double code(double x) {
	return hypot(x, x);
}

Java

public static double code(double x) {
	return Math.hypot(x, x);
}

Python

def code(x):
	return math.hypot(x, x)

Julia

function code(x)
	return hypot(x, x)
end

MATLAB

function tmp = code(x)
	tmp = hypot(x, x);
end

Wolfram

code[x_] := N[Sqrt[x ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 60.4%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 60.4%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{hypot}\left(x, x\right) \]

Alternative 2: 6.4% accurate, 101.7× speedup

Math

\[\begin{array}{l} \\ x + -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x -2.0))

C

double code(double x) {
	return x + -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (-2.0d0)
end function

Java

public static double code(double x) {
	return x + -2.0;
}

Python

def code(x):
	return x + -2.0

Julia

function code(x)
	return Float64(x + -2.0)
end

MATLAB

function tmp = code(x)
	tmp = x + -2.0;
end

Wolfram

code[x_] := N[(x + -2.0), $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 60.4%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 60.4%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
4. Step-by-step derivation

   1. add-cube-cbrt 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}} \]

   2. pow3 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right)}^{3}} \]

5. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right)}^{3}} \]
6. Step-by-step derivation

   1. unpow3 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}} \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

   3. hypot-udef 60.4%

      \[\leadsto \color{blue}{\sqrt{x \cdot x + x \cdot x}} \]

   4. count-2 60.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(x \cdot x\right)}} \]

   5. add-sqr-sqrt 59.9%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \left(x \cdot x\right)} \]

   6. swap-sqr 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot x\right) \cdot \left(\sqrt{2} \cdot x\right)}} \]

   7. *-commutative 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot \sqrt{2}\right)} \cdot \left(\sqrt{2} \cdot x\right)} \]

   8. *-commutative 60.1%

      \[\leadsto \sqrt{\left(x \cdot \sqrt{2}\right) \cdot \color{blue}{\left(x \cdot \sqrt{2}\right)}} \]

   9. sqrt-unprod 52.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot \sqrt{2}} \cdot \sqrt{x \cdot \sqrt{2}}} \]

   10. add-sqr-sqrt 53.8%

       \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]

   11. *-commutative 53.8%

       \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

   12. add-sqr-sqrt 53.7%

       \[\leadsto \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot x \]

   13. associate-*l* 53.8%

       \[\leadsto \color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right)} \]

   14. pow1/2 53.8%

       \[\leadsto \sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   15. sqrt-pow1 53.8%

       \[\leadsto \color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   16. metadata-eval 53.8%

       \[\leadsto {2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   17. pow1/2 53.8%

       \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   18. sqrt-pow1 53.8%

       \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   19. metadata-eval 53.8%

       \[\leadsto {2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot x\right) \]

7. Applied egg-rr 53.8%

   \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]
8. Step-by-step derivation

   1. associate-*r* 53.7%

      \[\leadsto \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot x} \]

   2. add-sqr-sqrt 52.5%

      \[\leadsto \left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

   3. associate-*r* 52.6%

      \[\leadsto \color{blue}{\left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \sqrt{x}\right) \cdot \sqrt{x}} \]

   4. pow-prod-up 52.7%

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

   5. metadata-eval 52.7%

      \[\leadsto \left({2}^{\color{blue}{0.5}} \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

   6. pow1/2 52.7%

      \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

   7. sqrt-prod 52.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot x}} \cdot \sqrt{x} \]

   8. count-2 52.9%

      \[\leadsto \sqrt{\color{blue}{x + x}} \cdot \sqrt{x} \]

   9. sqrt-prod 60.4%

      \[\leadsto \color{blue}{\sqrt{\left(x + x\right) \cdot x}} \]

   10. *-commutative 60.4%

       \[\leadsto \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]

   11. flip-+ 0.0%

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

   12. +-inverses 0.0%

       \[\leadsto \sqrt{x \cdot \frac{\color{blue}{0}}{x - x}} \]

   13. +-inverses 0.0%

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

   14. associate-*r/ 0.0%

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

   15. distribute-lft-out-- 0.0%

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

9. Applied egg-rr 11.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, x\right)} \]

11. Simplified 6.7%

    \[\leadsto \color{blue}{-2 + x} \]

12. Final simplification 6.7%

    \[\leadsto x + -2 \]

Alternative 3: 11.3% accurate, 101.7× speedup

Math

\[\begin{array}{l} \\ x + x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x x))

C

double code(double x) {
	return x + x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + x
end function

Java

public static double code(double x) {
	return x + x;
}

Python

def code(x):
	return x + x

Julia

function code(x)
	return Float64(x + x)
end

MATLAB

function tmp = code(x)
	tmp = x + x;
end

Wolfram

code[x_] := N[(x + x), $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 60.4%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 60.4%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
4. Step-by-step derivation

   1. add-cube-cbrt 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}} \]

   2. pow3 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right)}^{3}} \]

5. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right)}^{3}} \]
6. Step-by-step derivation

   1. unpow3 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}} \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

   3. hypot-udef 60.4%

      \[\leadsto \color{blue}{\sqrt{x \cdot x + x \cdot x}} \]

   4. count-2 60.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(x \cdot x\right)}} \]

   5. add-sqr-sqrt 59.9%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \left(x \cdot x\right)} \]

   6. swap-sqr 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot x\right) \cdot \left(\sqrt{2} \cdot x\right)}} \]

   7. *-commutative 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot \sqrt{2}\right)} \cdot \left(\sqrt{2} \cdot x\right)} \]

   8. *-commutative 60.1%

      \[\leadsto \sqrt{\left(x \cdot \sqrt{2}\right) \cdot \color{blue}{\left(x \cdot \sqrt{2}\right)}} \]

   9. sqrt-unprod 52.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot \sqrt{2}} \cdot \sqrt{x \cdot \sqrt{2}}} \]

   10. add-sqr-sqrt 53.8%

       \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]

   11. *-commutative 53.8%

       \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

   12. add-sqr-sqrt 53.7%

       \[\leadsto \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot x \]

   13. associate-*l* 53.8%

       \[\leadsto \color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right)} \]

   14. pow1/2 53.8%

       \[\leadsto \sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   15. sqrt-pow1 53.8%

       \[\leadsto \color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   16. metadata-eval 53.8%

       \[\leadsto {2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   17. pow1/2 53.8%

       \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   18. sqrt-pow1 53.8%

       \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   19. metadata-eval 53.8%

       \[\leadsto {2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot x\right) \]

7. Applied egg-rr 53.8%

   \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]
8. Step-by-step derivation

   1. associate-*r* 53.7%

      \[\leadsto \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot x} \]

   2. add-sqr-sqrt 52.5%

      \[\leadsto \left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

   3. associate-*r* 52.6%

      \[\leadsto \color{blue}{\left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \sqrt{x}\right) \cdot \sqrt{x}} \]

   4. pow-prod-up 52.7%

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

   5. metadata-eval 52.7%

      \[\leadsto \left({2}^{\color{blue}{0.5}} \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

   6. pow1/2 52.7%

      \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

   7. sqrt-prod 52.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot x}} \cdot \sqrt{x} \]

   8. count-2 52.9%

      \[\leadsto \sqrt{\color{blue}{x + x}} \cdot \sqrt{x} \]

   9. sqrt-prod 60.4%

      \[\leadsto \color{blue}{\sqrt{\left(x + x\right) \cdot x}} \]

   10. *-commutative 60.4%

       \[\leadsto \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]

   11. flip-+ 0.0%

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

   12. +-inverses 0.0%

       \[\leadsto \sqrt{x \cdot \frac{\color{blue}{0}}{x - x}} \]

   13. +-inverses 0.0%

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

   14. associate-*r/ 0.0%

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

   15. distribute-lft-out-- 0.0%

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

9. Applied egg-rr 11.8%

   \[\leadsto \color{blue}{x + x} \]

10. Final simplification 11.8%

    \[\leadsto x + x \]

Alternative 4: 1.7% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 60.4%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 60.4%

      \[\leadsto \sqrt{\color{blue}{x \cdot x} + {x}^{2}} \]

   2. unpow2 60.4%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{x \cdot x}} \]

   3. hypot-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
4. Step-by-step derivation

   1. add-cube-cbrt 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}} \]

   2. pow3 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right)}^{3}} \]

5. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right)}^{3}} \]
6. Step-by-step derivation

   1. unpow3 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(x, x\right)}} \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

   3. hypot-udef 60.4%

      \[\leadsto \color{blue}{\sqrt{x \cdot x + x \cdot x}} \]

   4. count-2 60.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(x \cdot x\right)}} \]

   5. add-sqr-sqrt 59.9%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \left(x \cdot x\right)} \]

   6. swap-sqr 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot x\right) \cdot \left(\sqrt{2} \cdot x\right)}} \]

   7. *-commutative 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot \sqrt{2}\right)} \cdot \left(\sqrt{2} \cdot x\right)} \]

   8. *-commutative 60.1%

      \[\leadsto \sqrt{\left(x \cdot \sqrt{2}\right) \cdot \color{blue}{\left(x \cdot \sqrt{2}\right)}} \]

   9. sqrt-unprod 52.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot \sqrt{2}} \cdot \sqrt{x \cdot \sqrt{2}}} \]

   10. add-sqr-sqrt 53.8%

       \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]

   11. *-commutative 53.8%

       \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

   12. add-sqr-sqrt 53.7%

       \[\leadsto \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot x \]

   13. associate-*l* 53.8%

       \[\leadsto \color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right)} \]

   14. pow1/2 53.8%

       \[\leadsto \sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   15. sqrt-pow1 53.8%

       \[\leadsto \color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   16. metadata-eval 53.8%

       \[\leadsto {2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   17. pow1/2 53.8%

       \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   18. sqrt-pow1 53.8%

       \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   19. metadata-eval 53.8%

       \[\leadsto {2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot x\right) \]

7. Applied egg-rr 53.8%

   \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]

8. Taylor expanded in x around 0 53.8%

   \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]

10. Simplified 1.6%

    \[\leadsto \color{blue}{-2} \]

11. Final simplification 1.6%

    \[\leadsto -2 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))