sqrt E (should all be same)

Percentage Accurate: 54.1% → 100.0%

Time: 4.3s

Alternatives: 3

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: 54.1% 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, 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 56.7%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-hypot.f64 100.0%

      \[\leadsto \mathsf{hypot.f64}\left(x, \color{blue}{x}\right) \]

4. Applied egg-rr 100.0%

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

Alternative 2: 50.3% 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 56.7%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{{\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}}{{x}^{2} \cdot {x}^{2} + \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{{x}^{2} \cdot {x}^{2} + \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)}{{\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}}}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{2}{2}}{\frac{{x}^{2} \cdot {x}^{2} + \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)}{{\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}}}\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{2}\right), \left(\frac{{x}^{2} \cdot {x}^{2} + \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)}{{\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{x}^{2} \cdot {x}^{2} + \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)}{{\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({x}^{2} \cdot {x}^{2} + \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)\right), \left({\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right), \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)\right), \left({\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right), \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)\right), \left({\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right), \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)\right), \left({\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right), \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)\right), \left({\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)\right), \left({\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}\right)\right)\right)\right) \]

   12. +-inverses N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right), 0\right), \left({\left({x}^{2}\right)}^{3} + {\left({x}^{2}\right)}^{3}\right)\right)\right)\right) \]

   13. sum-cubes N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right), 0\right), \left(\left({x}^{2} \cdot {x}^{2} + \left({x}^{2} \cdot {x}^{2} - {x}^{2} \cdot {x}^{2}\right)\right) \cdot \left({x}^{2} + {x}^{2}\right)\right)\right)\right)\right) \]

4. Applied egg-rr 15.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2}}\right)}\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right) \]

   2. +-rgt-identity N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2} + 0\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x + 0\right)\right)\right)\right) \]

   4. accelerator-lowering-fma.f64 56.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(x, x, 0\right)\right)\right)\right) \]

7. Simplified 56.6%

   \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{0.5}{\mathsf{fma}\left(x, x, 0\right)}}}} \]
8. Step-by-step derivation

   1. pow1/2 N/A

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

   2. +-rgt-identity N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. unpow-prod-down N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. unpow1 N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}}\right), \color{blue}{x}\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), x\right) \]

   12. sqrt-lowering-sqrt.f64 54.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), x\right) \]

9. Applied egg-rr 54.2%

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

10. Final simplification 54.2%

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

Alternative 3: 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 56.7%

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

   1. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(x \cdot x + {x}^{2}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(x \cdot x + x \cdot x\right)\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(x \cdot \left(x + x\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(x + x\right)\right)\right) \]

   5. +-lowering-+.f64 56.7%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, x\right)\right)\right) \]

4. Applied egg-rr 56.7%

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

   1. distribute-lft-in N/A

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

   2. pow1/2 N/A

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

   3. flip-+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. flip-+ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 3.8%

       \[\leadsto 0 \]

6. Applied egg-rr 3.8%

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))