sqrt E (should all be same)

Percentage Accurate: 53.7% → 100.0%

Time: 2.4s

Alternatives: 6

Speedup: 9.0×

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: 53.7% 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, 2.1× 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 54.1%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-hypot.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + x} \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310) (- (* x (sqrt 2.0))) (* (sqrt (+ x x)) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = sqrt((x + x)) * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -(x * sqrt(2.0d0))
    else
        tmp = sqrt((x + x)) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -(x * Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt((x + x)) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -(x * math.sqrt(2.0))
	else:
		tmp = math.sqrt((x + x)) * math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(-Float64(x * sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(x + x)) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -(x * sqrt(2.0));
	else
		tmp = sqrt((x + x)) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], (-N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[N[(x + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 52.0%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 99.3

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

   5. Simplified 99.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 56.5%

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 56.5

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

   4. Applied egg-rr 56.5%

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(x \cdot \left(x + x\right)\right)}^{\frac{1}{2}}} \]

      2. unpow-prod-down N/A

         \[\leadsto \color{blue}{{x}^{\frac{1}{2}} \cdot {\left(x + x\right)}^{\frac{1}{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{\left(x + x\right)}^{\frac{1}{2}} \cdot {x}^{\frac{1}{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(x + x\right)}^{\frac{1}{2}} \cdot {x}^{\frac{1}{2}}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\sqrt{x + x} \cdot \sqrt{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (sqrt 2.0)))) (if (<= x -5e-310) (- t_0) t_0)))

C

double code(double x) {
	double t_0 = x * sqrt(2.0);
	double tmp;
	if (x <= -5e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * sqrt(2.0d0)
    if (x <= (-5d-310)) then
        tmp = -t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * Math.sqrt(2.0);
	double tmp;
	if (x <= -5e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * math.sqrt(2.0)
	tmp = 0
	if x <= -5e-310:
		tmp = -t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(-t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * sqrt(2.0);
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-310], (-t$95$0), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 52.0%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 99.3

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

   5. Simplified 99.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 56.5%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 99.3

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 5.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.95 \cdot 10^{-206}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.95e-206) 0.0 (sqrt (+ x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.95e-206) {
		tmp = 0.0;
	} else {
		tmp = sqrt((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.95d-206) then
        tmp = 0.0d0
    else
        tmp = sqrt((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.95e-206) {
		tmp = 0.0;
	} else {
		tmp = Math.sqrt((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.95e-206:
		tmp = 0.0
	else:
		tmp = math.sqrt((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.95e-206)
		tmp = 0.0;
	else
		tmp = sqrt(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.95e-206)
		tmp = 0.0;
	else
		tmp = sqrt((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.95e-206], 0.0, N[Sqrt[N[(x + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9500000000000001e-206

   1. Initial program 47.0%

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 47.0

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

   4. Applied egg-rr 47.0%

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]
   5. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. +-inverses N/A

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

      6. +-inverses N/A

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

      7. +-inverses N/A

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

      8. flip-+ N/A

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

      9. +-lowering-+.f64 0.6

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

   6. Applied egg-rr 0.6%

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

      1. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(x + x\right)}^{\frac{1}{2}}} \]

      2. flip-+ N/A

         \[\leadsto {\color{blue}{\left(\frac{x \cdot x - x \cdot x}{x - x}\right)}}^{\frac{1}{2}} \]

      3. +-inverses N/A

         \[\leadsto {\left(\frac{\color{blue}{0}}{x - x}\right)}^{\frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto {\left(\frac{\color{blue}{0 - 0}}{x - x}\right)}^{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto {\left(\frac{\color{blue}{0 \cdot 0} - 0}{x - x}\right)}^{\frac{1}{2}} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{x - x}\right)}^{\frac{1}{2}} \]

      7. +-inverses N/A

         \[\leadsto {\left(\frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right)}^{\frac{1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto {\left(\frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right)}^{\frac{1}{2}} \]

      9. flip-- N/A

         \[\leadsto {\color{blue}{\left(0 - 0\right)}}^{\frac{1}{2}} \]

      10. metadata-eval N/A

          \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

      11. metadata-eval 3.8

          \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 3.8%

      \[\leadsto \color{blue}{0} \]

   if 3.9500000000000001e-206 < x

   1. Initial program 65.4%

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 65.4

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

   4. Applied egg-rr 65.4%

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]
   5. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. +-inverses N/A

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

      6. +-inverses N/A

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

      7. +-inverses N/A

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

      8. flip-+ N/A

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

      9. +-lowering-+.f64 7.2

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

   6. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\sqrt{x + x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 51.3% accurate, 13.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (sqrt 2.0)))

C

double code(double x) {
	return x * sqrt(2.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * Math.sqrt(2.0);
}

Python

def code(x):
	return x * math.sqrt(2.0)

Julia

function code(x)
	return Float64(x * sqrt(2.0))
end

MATLAB

function tmp = code(x)
	tmp = x * sqrt(2.0);
end

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 45.9

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

5. Simplified 45.9%

   \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
6. Add Preprocessing

Alternative 6: 3.8% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 54.1%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. distribute-lft-out N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-lowering-+.f64 54.1

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

4. Applied egg-rr 54.1%

   \[\leadsto \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]
5. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. flip-+ N/A

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

   4. +-inverses N/A

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

   5. +-inverses N/A

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

   6. +-inverses N/A

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

   7. +-inverses N/A

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

   8. flip-+ N/A

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

   9. +-lowering-+.f64 3.1

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

6. Applied egg-rr 3.1%

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

   1. pow1/2 N/A

      \[\leadsto \color{blue}{{\left(x + x\right)}^{\frac{1}{2}}} \]

   2. flip-+ N/A

      \[\leadsto {\color{blue}{\left(\frac{x \cdot x - x \cdot x}{x - x}\right)}}^{\frac{1}{2}} \]

   3. +-inverses N/A

      \[\leadsto {\left(\frac{\color{blue}{0}}{x - x}\right)}^{\frac{1}{2}} \]

   4. metadata-eval N/A

      \[\leadsto {\left(\frac{\color{blue}{0 - 0}}{x - x}\right)}^{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto {\left(\frac{\color{blue}{0 \cdot 0} - 0}{x - x}\right)}^{\frac{1}{2}} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{x - x}\right)}^{\frac{1}{2}} \]

   7. +-inverses N/A

      \[\leadsto {\left(\frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right)}^{\frac{1}{2}} \]

   8. metadata-eval N/A

      \[\leadsto {\left(\frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right)}^{\frac{1}{2}} \]

   9. flip-- N/A

      \[\leadsto {\color{blue}{\left(0 - 0\right)}}^{\frac{1}{2}} \]

   10. metadata-eval N/A

       \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

   11. metadata-eval 3.5

       \[\leadsto \color{blue}{0} \]

8. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))