sqrt C (should all be same)

Percentage Accurate: 55.0% → 99.4%

Time: 802.0ms

Alternatives: 3

Speedup: 1.2×

Specification

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (sqrt (* 2 (* x x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 55.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (sqrt (* 2 (* x x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\sqrt{\left|x + x\right|} \cdot \sqrt{\left|x\right|} \]

FPCore

(FPCore (x)
  :precision binary64
  (* (sqrt (fabs (+ x x))) (sqrt (fabs x))))

C

double code(double x) {
	return sqrt(fabs((x + x))) * sqrt(fabs(x));
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = sqrt(abs((x + x))) * sqrt(abs(x))
end function

Java

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

Python

def code(x):
	return math.sqrt(math.fabs((x + x))) * math.sqrt(math.fabs(x))

Julia

function code(x)
	return Float64(sqrt(abs(Float64(x + x))) * sqrt(abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(abs((x + x))) * sqrt(abs(x));
end

Wolfram

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

Derivation

1. Initial program 55.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. metadata-eval N/A

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

   7. pow-plus-rev N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. count-2-rev N/A

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

   11. count-2-rev N/A

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

   12. sqrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-fabs.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lower-fabs.f64 N/A

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

   19. count-2-rev N/A

       \[\leadsto \sqrt{\left|{x}^{1}\right|} \cdot \sqrt{\left|\color{blue}{x + x}\right|} \]

   20. lower-+.f64 99.4%

       \[\leadsto \sqrt{\left|{x}^{1}\right|} \cdot \sqrt{\left|\color{blue}{x + x}\right|} \]

3. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\left|{x}^{1}\right|} \cdot \sqrt{\left|x + x\right|}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\left|{x}^{1}\right|} \cdot \sqrt{\left|x + x\right|}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\left|x + x\right|} \cdot \sqrt{\left|{x}^{1}\right|}} \]

   3. lower-*.f64 99.4%

      \[\leadsto \color{blue}{\sqrt{\left|x + x\right|} \cdot \sqrt{\left|{x}^{1}\right|}} \]

   4. lift-fabs.f64 N/A

      \[\leadsto \sqrt{\left|x + x\right|} \cdot \sqrt{\color{blue}{\left|{x}^{1}\right|}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \sqrt{\left|x + x\right|} \cdot \sqrt{\left|\color{blue}{{x}^{1}}\right|} \]

   6. unpow1 N/A

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

   7. lift-fabs.f64 99.4%

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\left|x + x\right|} \cdot \sqrt{\left|x\right|}} \]
6. Add Preprocessing

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\sqrt{2} \cdot \left|x\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (* (sqrt 2) (fabs x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[2], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lift-*.f64 N/A

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

   5. fabs-sqr N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-fabs.f64 99.3%

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

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left|x\right|} \]
4. Add Preprocessing

Alternative 3: 7.0% accurate, 1.3× speedup

Math

\[\sqrt{\left|x + x\right|} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = sqrt(abs((x + x)))
end function

Java

public static double code(double x) {
	return Math.sqrt(Math.abs((x + x)));
}

Python

def code(x):
	return math.sqrt(math.fabs((x + x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. rem-square-sqrt N/A

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

   4. sqrt-unprod N/A

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

   5. rem-square-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. flip-+ N/A

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

   8. +-inverses N/A

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

   9. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{\frac{0}{\color{blue}{\left|x \cdot x\right|} - \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\frac{0}{\left|\color{blue}{x \cdot x}\right| - \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}} \]

   11. mul-fabs N/A

       \[\leadsto \sqrt{\frac{0}{\color{blue}{\left|x\right| \cdot \left|x\right|} - \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}} \]

   12. sqr-abs-rev N/A

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

   13. lift-*.f64 N/A

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

   14. rem-sqrt-square-rev N/A

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

   15. lift-*.f64 N/A

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

   16. mul-fabs N/A

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

   17. sqr-abs-rev N/A

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

   18. lift-*.f64 N/A

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

   19. +-inverses N/A

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

   20. lower-/.f32 0.0%

       \[\leadsto \sqrt{\color{blue}{\left( \color{blue}{\frac{0}{0}} \right)_{\text{binary32}}}} \]

   21. lower-/.f32 N/A

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

   22. +-inverses N/A

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

   23. +-inverses N/A

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

3. Applied rewrites 7.0%

   \[\leadsto \sqrt{\color{blue}{\left|x + x\right|}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025326 -o generate:taylor -o generate:evaluate
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2 (* x x))))