sqrt D (should all be same)

Percentage Accurate: 54.7% → 100.0%

Time: 2.9s

Alternatives: 4

Speedup: 4.9×

Specification

Math

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

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: 54.7% accurate, 1.0× speedup

Math

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

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, 1.1× 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 52.8%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Add Preprocessing
3. 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

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

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
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -x * sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * 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(2.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x * math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * 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(2.0) * 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(2.0) * 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[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 54.3%

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

   5. Applied rewrites 99.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 51.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing
   3. 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. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 99.3

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

   4. Applied rewrites 99.3%

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

Alternative 3: 51.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-206) (sqrt 2.0) (* (sqrt 2.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -4e-206) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-4d-206)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e-206) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e-206:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-206)
		tmp = sqrt(2.0);
	else
		tmp = Float64(sqrt(2.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-206)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e-206], N[Sqrt[2.0], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000011e-206

   1. Initial program 60.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 60.9

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

   4. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1 - 1}}{x \cdot x - x \cdot x}} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot 1} - 1}{x \cdot x - x \cdot x}} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-inverses N/A

          \[\leadsto \sqrt{\frac{\frac{2}{2} \cdot \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}}{\color{blue}{0}}} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\frac{2}{2} \cdot \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}}{\color{blue}{1 - 1}}} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\frac{2}{2} \cdot \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}}{\color{blue}{\frac{2}{2}} - 1}} \]

      16. metadata-eval N/A

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

      17. flip-+ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{2}{2} + \frac{2}{2}}} \]

      18. div-add-rev N/A

          \[\leadsto \sqrt{\color{blue}{\frac{2 + 2}{2}}} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\color{blue}{4}}{2}} \]

      20. metadata-eval 5.4

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

   6. Applied rewrites 5.4%

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

   if -4.00000000000000011e-206 < x

   1. Initial program 46.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing
   3. 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. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 88.7

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

   4. Applied rewrites 88.7%

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

Alternative 4: 5.4% accurate, 10.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return sqrt(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)
end function

Java

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

Python

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

Julia

function code(x)
	return sqrt(2.0)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[2.0], $MachinePrecision]

Derivation

1. Initial program 52.8%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. lower-+.f64 52.8

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

4. Applied rewrites 52.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\color{blue}{1 - 1}}{x \cdot x - x \cdot x}} \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot 1} - 1}{x \cdot x - x \cdot x}} \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. +-inverses N/A

       \[\leadsto \sqrt{\frac{\frac{2}{2} \cdot \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}}{\color{blue}{0}}} \]

   14. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\frac{2}{2} \cdot \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}}{\color{blue}{1 - 1}}} \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\frac{2}{2} \cdot \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}}{\color{blue}{\frac{2}{2}} - 1}} \]

   16. metadata-eval N/A

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

   17. flip-+ N/A

       \[\leadsto \sqrt{\color{blue}{\frac{2}{2} + \frac{2}{2}}} \]

   18. div-add-rev N/A

       \[\leadsto \sqrt{\color{blue}{\frac{2 + 2}{2}}} \]

   19. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\color{blue}{4}}{2}} \]

   20. metadata-eval 5.2

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

6. Applied rewrites 5.2%

   \[\leadsto \sqrt{\color{blue}{2}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))