sqrt D (should all be same)

Percentage Accurate: 54.2% → 100.0%

Time: 1.9s

Alternatives: 3

Speedup: 4.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

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 54.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. sqrt-prod N/A

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

   8. *-commutative N/A

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

   9. lower-unsound-*.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-unsound-sqrt.f64 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f64 50.7%

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

3. Applied rewrites 50.7%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x + x}} \]
4. Add Preprocessing

Alternative 3: 99.3% accurate, 4.6× speedup

Math

\[\left|x\right| \cdot 1.4142135623730951 \]

FPCore

(FPCore (x)
  :precision binary64
  (* (fabs x) 1.4142135623730951))

C

double code(double x) {
	return fabs(x) * 1.4142135623730951;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = abs(x) * 1.4142135623730951d0
end function

Java

public static double code(double x) {
	return Math.abs(x) * 1.4142135623730951;
}

Python

def code(x):
	return math.fabs(x) * 1.4142135623730951

Julia

function code(x)
	return Float64(abs(x) * 1.4142135623730951)
end

MATLAB

function tmp = code(x)
	tmp = abs(x) * 1.4142135623730951;
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * 1.4142135623730951), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 51.7%

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

4. Applied rewrites 51.7%

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

5. Evaluated real constant 51.7%

   \[\leadsto x \cdot 1.4142135623730951 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025231 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))