sqrt E (should all be same)

Percentage Accurate: 54.9% → 100.0%

Time: 6.6s

Alternatives: 6

Speedup: 2.1×

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: 54.9% 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 51.9%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
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: 50.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{2}}\\ t_1 := \sqrt{t\_0}\\ \left(t\_0 \cdot t\_1\right) \cdot \left(x \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt 2.0))) (t_1 (sqrt t_0))) (* (* t_0 t_1) (* x t_1))))

C

double code(double x) {
	double t_0 = sqrt(sqrt(2.0));
	double t_1 = sqrt(t_0);
	return (t_0 * t_1) * (x * t_1);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sqrt(sqrt(2.0d0))
    t_1 = sqrt(t_0)
    code = (t_0 * t_1) * (x * t_1)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(Math.sqrt(2.0));
	double t_1 = Math.sqrt(t_0);
	return (t_0 * t_1) * (x * t_1);
}

Python

def code(x):
	t_0 = math.sqrt(math.sqrt(2.0))
	t_1 = math.sqrt(t_0)
	return (t_0 * t_1) * (x * t_1)

Julia

function code(x)
	t_0 = sqrt(sqrt(2.0))
	t_1 = sqrt(t_0)
	return Float64(Float64(t_0 * t_1) * Float64(x * t_1))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt(sqrt(2.0));
	t_1 = sqrt(t_0);
	tmp = (t_0 * t_1) * (x * t_1);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[2.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \sqrt{2} + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]

   3. sqrt-lowering-sqrt.f64 47.3

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\sqrt{2}}, 0\right) \]

5. Simplified 47.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]
6. Step-by-step derivation

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

   5. pow-lowering-pow.f64 N/A

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

   6. metadata-eval 47.3

      \[\leadsto \mathsf{fma}\left(x, {\left({2}^{\color{blue}{0.25}}\right)}^{2}, 0\right) \]

7. Applied egg-rr 47.3%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{{\left({2}^{0.25}\right)}^{2}}, 0\right) \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot {\left({2}^{\frac{1}{4}}\right)}^{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{\left({2}^{\frac{1}{4}}\right)}^{2} \cdot x} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)} \cdot x \]

   4. sqr-pow N/A

      \[\leadsto \left({2}^{\frac{1}{4}} \cdot \color{blue}{\left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)}\right) \cdot x \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)} \cdot x \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot x\right)} \]

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

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot x\right)} \]

9. Applied egg-rr 47.5%

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

10. Final simplification 47.5%

    \[\leadsto \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \cdot \left(x \cdot \sqrt{\sqrt{\sqrt{2}}}\right) \]
11. Add Preprocessing

Alternative 3: 50.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \sqrt{2} + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]

   3. sqrt-lowering-sqrt.f64 47.3

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\sqrt{2}}, 0\right) \]

5. Simplified 47.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]
6. Step-by-step derivation

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

   5. pow-lowering-pow.f64 N/A

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

   6. metadata-eval 47.3

      \[\leadsto \mathsf{fma}\left(x, {\left({2}^{\color{blue}{0.25}}\right)}^{2}, 0\right) \]

7. Applied egg-rr 47.3%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{{\left({2}^{0.25}\right)}^{2}}, 0\right) \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot {\left({2}^{\frac{1}{4}}\right)}^{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{\left({2}^{\frac{1}{4}}\right)}^{2} \cdot x} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)} \cdot x \]

   4. sqr-pow N/A

      \[\leadsto \left({2}^{\frac{1}{4}} \cdot \color{blue}{\left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)}\right) \cdot x \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)} \cdot x \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot x\right)} \]

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

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot x\right)} \]

9. Applied egg-rr 47.5%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. pow1/2 N/A

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

    4. pow-plus N/A

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

    5. pow1/2 N/A

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

    6. pow-prod-up N/A

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

    7. metadata-eval N/A

       \[\leadsto {\left(\sqrt{\sqrt{2}}\right)}^{\left(\color{blue}{\frac{3}{2}} + \frac{1}{2}\right)} \cdot x \]

    8. metadata-eval N/A

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

    9. pow2 N/A

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

    10. rem-square-sqrt N/A

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

    11. +-rgt-identity N/A

        \[\leadsto \color{blue}{\sqrt{2} \cdot x + 0} \]

    12. unpow1 N/A

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

    13. metadata-eval N/A

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

    14. pow-prod-up N/A

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

    15. inv-pow N/A

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

    16. +-rgt-identity N/A

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

    17. unpow-prod-down N/A

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

    18. pow2 N/A

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

    19. rem-square-sqrt N/A

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

    20. pow2 N/A

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

11. Applied egg-rr 47.4%

    \[\leadsto \color{blue}{\frac{2}{\frac{\sqrt{2}}{x}}} \]
12. Add Preprocessing

Alternative 4: 50.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \sqrt{2} + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]

   3. sqrt-lowering-sqrt.f64 47.3

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\sqrt{2}}, 0\right) \]

5. Simplified 47.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]
6. Step-by-step derivation

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

   5. pow-lowering-pow.f64 N/A

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

   6. metadata-eval 47.3

      \[\leadsto \mathsf{fma}\left(x, {\left({2}^{\color{blue}{0.25}}\right)}^{2}, 0\right) \]

7. Applied egg-rr 47.3%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{{\left({2}^{0.25}\right)}^{2}}, 0\right) \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot {\left({2}^{\frac{1}{4}}\right)}^{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{\left({2}^{\frac{1}{4}}\right)}^{2} \cdot x} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)} \cdot x \]

   4. sqr-pow N/A

      \[\leadsto \left({2}^{\frac{1}{4}} \cdot \color{blue}{\left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)}\right) \cdot x \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)} \cdot x \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot x\right)} \]

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

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\left(\frac{\frac{1}{4}}{2}\right)}\right) \cdot \left({2}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot x\right)} \]

9. Applied egg-rr 47.5%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. pow1/2 N/A

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

    4. pow-plus N/A

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

    5. pow1/2 N/A

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

    6. pow-prod-up N/A

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

    7. metadata-eval N/A

       \[\leadsto {\left(\sqrt{\sqrt{2}}\right)}^{\left(\color{blue}{\frac{3}{2}} + \frac{1}{2}\right)} \cdot x \]

    8. metadata-eval N/A

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

    9. pow2 N/A

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

    10. rem-square-sqrt N/A

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

    11. +-rgt-identity N/A

        \[\leadsto \color{blue}{\sqrt{2} \cdot x + 0} \]

    12. unpow1 N/A

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

    13. metadata-eval N/A

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

    14. pow-prod-up N/A

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

11. Applied egg-rr 47.4%

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

12. Final simplification 47.4%

    \[\leadsto 2 \cdot \frac{x}{\sqrt{2}} \]
13. Add Preprocessing

Alternative 5: 50.7% 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 51.9%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \sqrt{2} + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]

   3. sqrt-lowering-sqrt.f64 47.3

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\sqrt{2}}, 0\right) \]

5. Simplified 47.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sqrt{2}, 0\right)} \]
6. Step-by-step derivation

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

   5. pow-lowering-pow.f64 N/A

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

   6. metadata-eval 47.3

      \[\leadsto \mathsf{fma}\left(x, {\left({2}^{\color{blue}{0.25}}\right)}^{2}, 0\right) \]

7. Applied egg-rr 47.3%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{{\left({2}^{0.25}\right)}^{2}}, 0\right) \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot {\left({2}^{\frac{1}{4}}\right)}^{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{\left({2}^{\frac{1}{4}}\right)}^{2} \cdot x} \]

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

      \[\leadsto \color{blue}{{\left({2}^{\frac{1}{4}}\right)}^{2} \cdot x} \]

   4. pow-pow N/A

      \[\leadsto \color{blue}{{2}^{\left(\frac{1}{4} \cdot 2\right)}} \cdot x \]

   5. metadata-eval N/A

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

   6. unpow1/2 N/A

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

   7. sqrt-lowering-sqrt.f64 47.3

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

9. Applied egg-rr 47.3%

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

10. Final simplification 47.3%

    \[\leadsto x \cdot \sqrt{2} \]
11. 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 51.9%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
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 51.9

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

4. Applied egg-rr 51.9%

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

   1. distribute-lft-in N/A

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

   2. pow1/2 N/A

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

   3. flip-+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. flip-+ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 3.8

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

6. Applied egg-rr 3.8%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))