2isqrt (example 3.6)

Percentage Accurate: 38.2% → 98.9%

Time: 4.0s

Alternatives: 10

Speedup: 1.5×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

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 = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 38.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

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 = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;x \leq 260000000:\\ \;\;\;\;\frac{t\_0 - \sqrt{x}}{\sqrt{x} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= x 260000000.0)
     (/ (- t_0 (sqrt x)) (* (sqrt x) t_0))
     (* (pow x -1.5) 0.5))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (x <= 260000000.0) {
		tmp = (t_0 - sqrt(x)) / (sqrt(x) * t_0);
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (x <= 260000000.0d0) then
        tmp = (t_0 - sqrt(x)) / (sqrt(x) * t_0)
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (x <= 260000000.0) {
		tmp = (t_0 - Math.sqrt(x)) / (Math.sqrt(x) * t_0);
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if x <= 260000000.0:
		tmp = (t_0 - math.sqrt(x)) / (math.sqrt(x) * t_0)
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (x <= 260000000.0)
		tmp = Float64(Float64(t_0 - sqrt(x)) / Float64(sqrt(x) * t_0));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (x <= 260000000.0)
		tmp = (t_0 - sqrt(x)) / (sqrt(x) * t_0);
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 260000000.0], N[(N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6e8

   1. Initial program 79.2%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. frac-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-+.f64 81.8

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

   4. Applied rewrites 81.8%

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

   if 2.6e8 < x

   1. Initial program 35.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   7. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{x}^{3}}} \cdot \color{blue}{\frac{1}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{x}^{3}}} \cdot \color{blue}{\frac{1}{2}} \]

      7. pow-flip N/A

         \[\leadsto \sqrt{{x}^{\left(\mathsf{neg}\left(3\right)\right)}} \cdot \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{{x}^{-3}} \cdot \frac{1}{2} \]

      9. sqrt-pow1 N/A

         \[\leadsto {x}^{\left(\frac{-3}{2}\right)} \cdot \frac{1}{2} \]

      10. lower-pow.f64 N/A

          \[\leadsto {x}^{\left(\frac{-3}{2}\right)} \cdot \frac{1}{2} \]

      11. metadata-eval 99.4

          \[\leadsto {x}^{-1.5} \cdot 0.5 \]

   8. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 260000000:\\ \;\;\;\;\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x} \cdot \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, \frac{{x}^{-0.5} \cdot 0.375}{-x}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (pow x -0.5) 0.5 (/ (* (pow x -0.5) 0.375) (- x))) x))

C

double code(double x) {
	return fma(pow(x, -0.5), 0.5, ((pow(x, -0.5) * 0.375) / -x)) / x;
}

Julia

function code(x)
	return Float64(fma((x ^ -0.5), 0.5, Float64(Float64((x ^ -0.5) * 0.375) / Float64(-x))) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[x, -0.5], $MachinePrecision] * 0.5 + N[(N[(N[Power[x, -0.5], $MachinePrecision] * 0.375), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 84.3%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{\color{blue}{x}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

8. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{\color{blue}{x}} \]

9. Final simplification 98.1%

   \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, \frac{{x}^{-0.5} \cdot 0.375}{-x}\right)}{x} \]
10. Add Preprocessing

Alternative 3: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{x}^{-0.5} \cdot 0.375}{-x}}{x} + \frac{\frac{0.5}{\sqrt{x}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ (/ (* (pow x -0.5) 0.375) (- x)) x) (/ (/ 0.5 (sqrt x)) x)))

C

double code(double x) {
	return (((pow(x, -0.5) * 0.375) / -x) / x) + ((0.5 / sqrt(x)) / 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 = ((((x ** (-0.5d0)) * 0.375d0) / -x) / x) + ((0.5d0 / sqrt(x)) / x)
end function

Java

public static double code(double x) {
	return (((Math.pow(x, -0.5) * 0.375) / -x) / x) + ((0.5 / Math.sqrt(x)) / x);
}

Python

def code(x):
	return (((math.pow(x, -0.5) * 0.375) / -x) / x) + ((0.5 / math.sqrt(x)) / x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64((x ^ -0.5) * 0.375) / Float64(-x)) / x) + Float64(Float64(0.5 / sqrt(x)) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((((x ^ -0.5) * 0.375) / -x) / x) + ((0.5 / sqrt(x)) / x);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[Power[x, -0.5], $MachinePrecision] * 0.375), $MachinePrecision] / (-x)), $MachinePrecision] / x), $MachinePrecision] + N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 84.3%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{\color{blue}{x}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

8. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{\color{blue}{x}} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-1}{2}}, \frac{1}{2}, -\frac{{x}^{\frac{-1}{2}} \cdot \frac{3}{8}}{x}\right)}{x} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{-1}{2}\right)}, \frac{1}{2}, -\frac{{x}^{\frac{-1}{2}} \cdot \frac{3}{8}}{x}\right)}{x} \]

   3. sqrt-pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt{x}\right)}^{-1}, \frac{1}{2}, -\frac{{x}^{\frac{-1}{2}} \cdot \frac{3}{8}}{x}\right)}{x} \]

   4. inv-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{1}{2}, -\frac{{x}^{\frac{-1}{2}} \cdot \frac{3}{8}}{x}\right)}{x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{1}{2}, -\frac{{x}^{\frac{-1}{2}} \cdot \frac{3}{8}}{x}\right)}{x} \]

   6. lift-sqrt.f64 98.0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{x} \]

10. Applied rewrites 98.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{x} \]

12. Applied rewrites 98.0%

    \[\leadsto \frac{\frac{{x}^{-0.5} \cdot 0.375}{-x}}{x} + \frac{\frac{0.5}{\sqrt{x}}}{\color{blue}{x}} \]
13. Add Preprocessing

Alternative 4: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({x}^{-1.5}, -0.375, \frac{0.5}{\sqrt{x}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (pow x -1.5) -0.375 (/ 0.5 (sqrt x))) x))

C

double code(double x) {
	return fma(pow(x, -1.5), -0.375, (0.5 / sqrt(x))) / x;
}

Julia

function code(x)
	return Float64(fma((x ^ -1.5), -0.375, Float64(0.5 / sqrt(x))) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[x, -1.5], $MachinePrecision] * -0.375 + N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 84.3%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{\color{blue}{x}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

8. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{\color{blue}{x}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} - \frac{3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]
10. Step-by-step derivation

    1. fp-cancel-sub-sign-inv N/A

       \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    3. +-commutative N/A

       \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

    5. +-commutative N/A

       \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{\sqrt{{x}^{3}} \cdot \frac{1}{2} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\sqrt{{x}^{3}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    8. sqrt-pow1 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    9. lower-pow.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    10. metadata-eval N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    11. lower-*.f64 N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    12. lift-sqrt.f64 N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    13. pow2 N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

    14. lift-*.f64 55.2

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

11. Applied rewrites 55.2%

    \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{\frac{-3}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\sqrt{\frac{1}{{x}^{3}}} \cdot \frac{-3}{8} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

    2. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, \frac{-3}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    3. pow-flip N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{{x}^{\left(\mathsf{neg}\left(3\right)\right)}}, \frac{-3}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    4. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{{x}^{-3}}, \frac{-3}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    5. sqrt-pow1 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{-3}{2}\right)}, \frac{-3}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    6. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{-3}{2}\right)}, \frac{-3}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-3}{2}}, \frac{-3}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

    8. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-3}{2}}, \frac{-3}{8}, \sqrt{\frac{1}{x}} \cdot \frac{1}{2}\right)}{x} \]

    9. sqrt-div N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-3}{2}}, \frac{-3}{8}, \frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{1}{2}\right)}{x} \]

    10. metadata-eval N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-3}{2}}, \frac{-3}{8}, \frac{1}{\sqrt{x}} \cdot \frac{1}{2}\right)}{x} \]

    11. associate-*l/ N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-3}{2}}, \frac{-3}{8}, \frac{1 \cdot \frac{1}{2}}{\sqrt{x}}\right)}{x} \]

    12. metadata-eval N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-3}{2}}, \frac{-3}{8}, \frac{\frac{1}{2}}{\sqrt{x}}\right)}{x} \]

    13. lower-/.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-3}{2}}, \frac{-3}{8}, \frac{\frac{1}{2}}{\sqrt{x}}\right)}{x} \]

    14. lift-sqrt.f64 98.0

        \[\leadsto \frac{\mathsf{fma}\left({x}^{-1.5}, -0.375, \frac{0.5}{\sqrt{x}}\right)}{x} \]

14. Applied rewrites 98.0%

    \[\leadsto \frac{\mathsf{fma}\left({x}^{-1.5}, -0.375, \frac{0.5}{\sqrt{x}}\right)}{x} \]
15. Add Preprocessing

Alternative 5: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;x \leq 260000000:\\ \;\;\;\;\frac{t\_0 - \sqrt{x}}{\sqrt{x} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= x 260000000.0)
     (/ (- t_0 (sqrt x)) (* (sqrt x) t_0))
     (/ (/ 0.5 (sqrt x)) x))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (x <= 260000000.0) {
		tmp = (t_0 - sqrt(x)) / (sqrt(x) * t_0);
	} else {
		tmp = (0.5 / sqrt(x)) / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (x <= 260000000.0d0) then
        tmp = (t_0 - sqrt(x)) / (sqrt(x) * t_0)
    else
        tmp = (0.5d0 / sqrt(x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (x <= 260000000.0) {
		tmp = (t_0 - Math.sqrt(x)) / (Math.sqrt(x) * t_0);
	} else {
		tmp = (0.5 / Math.sqrt(x)) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if x <= 260000000.0:
		tmp = (t_0 - math.sqrt(x)) / (math.sqrt(x) * t_0)
	else:
		tmp = (0.5 / math.sqrt(x)) / x
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (x <= 260000000.0)
		tmp = Float64(Float64(t_0 - sqrt(x)) / Float64(sqrt(x) * t_0));
	else
		tmp = Float64(Float64(0.5 / sqrt(x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (x <= 260000000.0)
		tmp = (t_0 - sqrt(x)) / (sqrt(x) * t_0);
	else
		tmp = (0.5 / sqrt(x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 260000000.0], N[(N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6e8

   1. Initial program 79.2%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. frac-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-+.f64 81.8

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

   4. Applied rewrites 81.8%

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

   if 2.6e8 < x

   1. Initial program 35.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{\color{blue}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} - \frac{3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]
   10. Step-by-step derivation

       1. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{\sqrt{{x}^{3}} \cdot \frac{1}{2} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\sqrt{{x}^{3}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       8. sqrt-pow1 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       9. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       12. lift-sqrt.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       13. pow2 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

       14. lift-*.f64 55.0

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

   11. Applied rewrites 55.0%

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

   12. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{x} \]

       7. lift-sqrt.f64 99.0

          \[\leadsto \frac{\frac{0.5}{\sqrt{x}}}{x} \]

   14. Applied rewrites 99.0%

       \[\leadsto \frac{\frac{0.5}{\sqrt{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 260000000:\\ \;\;\;\;\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x} \cdot \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;\sqrt{\frac{1}{x}} - \frac{1}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 90000000.0)
   (- (sqrt (/ 1.0 x)) (/ 1.0 (sqrt (+ x 1.0))))
   (/ (/ 0.5 (sqrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = sqrt((1.0 / x)) - (1.0 / sqrt((x + 1.0)));
	} else {
		tmp = (0.5 / sqrt(x)) / 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 <= 90000000.0d0) then
        tmp = sqrt((1.0d0 / x)) - (1.0d0 / sqrt((x + 1.0d0)))
    else
        tmp = (0.5d0 / sqrt(x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = Math.sqrt((1.0 / x)) - (1.0 / Math.sqrt((x + 1.0)));
	} else {
		tmp = (0.5 / Math.sqrt(x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 90000000.0:
		tmp = math.sqrt((1.0 / x)) - (1.0 / math.sqrt((x + 1.0)))
	else:
		tmp = (0.5 / math.sqrt(x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 90000000.0)
		tmp = Float64(sqrt(Float64(1.0 / x)) - Float64(1.0 / sqrt(Float64(x + 1.0))));
	else
		tmp = Float64(Float64(0.5 / sqrt(x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 90000000.0)
		tmp = sqrt((1.0 / x)) - (1.0 / sqrt((x + 1.0)));
	else
		tmp = (0.5 / sqrt(x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 90000000.0], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e7

   1. Initial program 81.8%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-sqrt.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 81.9

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

   4. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\sqrt{{x}^{-1}}} - \frac{1}{\sqrt{x + 1}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-/.f64 81.9

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

   6. Applied rewrites 81.9%

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

   if 9e7 < x

   1. Initial program 35.1%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{\color{blue}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{\color{blue}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} - \frac{3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]
   10. Step-by-step derivation

       1. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{\sqrt{{x}^{3}} \cdot \frac{1}{2} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\sqrt{{x}^{3}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       8. sqrt-pow1 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       9. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       12. lift-sqrt.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

       13. pow2 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

       14. lift-*.f64 55.2

           \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

   11. Applied rewrites 55.2%

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

   12. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{x} \]

       7. lift-sqrt.f64 98.8

          \[\leadsto \frac{\frac{0.5}{\sqrt{x}}}{x} \]

   14. Applied rewrites 98.8%

       \[\leadsto \frac{\frac{0.5}{\sqrt{x}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{\sqrt{x}}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 (sqrt x)) x))

C

double code(double x) {
	return (0.5 / sqrt(x)) / 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 = (0.5d0 / sqrt(x)) / x
end function

Java

public static double code(double x) {
	return (0.5 / Math.sqrt(x)) / x;
}

Python

def code(x):
	return (0.5 / math.sqrt(x)) / x

Julia

function code(x)
	return Float64(Float64(0.5 / sqrt(x)) / x)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / sqrt(x)) / x;
end

Wolfram

code[x_] := N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 84.3%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{\color{blue}{x}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]

8. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.5}, 0.5, -\frac{{x}^{-0.5} \cdot 0.375}{x}\right)}{\color{blue}{x}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} - \frac{3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]
10. Step-by-step derivation

    1. fp-cancel-sub-sign-inv N/A

       \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    3. +-commutative N/A

       \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{{x}^{3}}}{{x}^{2}}}{x} \]

    5. +-commutative N/A

       \[\leadsto \frac{\frac{\frac{1}{2} \cdot \sqrt{{x}^{3}} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{\sqrt{{x}^{3}} \cdot \frac{1}{2} + \frac{-3}{8} \cdot \sqrt{x}}{{x}^{2}}}{x} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\sqrt{{x}^{3}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    8. sqrt-pow1 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    9. lower-pow.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\left(\frac{3}{2}\right)}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    10. metadata-eval N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    11. lower-*.f64 N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    12. lift-sqrt.f64 N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{{x}^{2}}}{x} \]

    13. pow2 N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{\frac{3}{2}}, \frac{1}{2}, \frac{-3}{8} \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

    14. lift-*.f64 55.2

        \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

11. Applied rewrites 55.2%

    \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{1.5}, 0.5, -0.375 \cdot \sqrt{x}\right)}{x \cdot x}}{x} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x} \]
13. Step-by-step derivation

    1. *-commutative N/A

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

    2. sqrt-div N/A

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

    3. metadata-eval N/A

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

    4. associate-*l/ N/A

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

    5. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{x} \]

    6. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{x} \]

    7. lift-sqrt.f64 96.7

       \[\leadsto \frac{\frac{0.5}{\sqrt{x}}}{x} \]

14. Applied rewrites 96.7%

    \[\leadsto \frac{\frac{0.5}{\sqrt{x}}}{x} \]
15. Add Preprocessing

Alternative 8: 81.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{x}}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.5 (sqrt x)) (* x x)))

C

double code(double x) {
	return (0.5 * sqrt(x)) / (x * 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 = (0.5d0 * sqrt(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (0.5 * Math.sqrt(x)) / (x * x);
}

Python

def code(x):
	return (0.5 * math.sqrt(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(0.5 * sqrt(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * sqrt(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 84.3%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{{x}^{-5}} \cdot -0.5\right) \cdot \mathsf{fma}\left(0.5, x, 1\right) - \mathsf{fma}\left(-0.5, \sqrt{x}, \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right) \cdot \sqrt{{x}^{-3}}, -0.5, 0.5 \cdot {\left(\sqrt{x}\right)}^{-1}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{1}{2} \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 82.8

      \[\leadsto \frac{0.5 \cdot \sqrt{x}}{x \cdot x} \]

8. Applied rewrites 82.8%

   \[\leadsto \frac{0.5 \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
9. Add Preprocessing

Alternative 9: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sqrt{x} + x \cdot \left(\frac{1}{2} \cdot x - 1\right)}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot x - 1, x, \sqrt{x}\right)}{x} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot x - 1, x, \sqrt{x}\right)}{x} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot x - 1, x, \sqrt{x}\right)}{x} \]

   7. lift-sqrt.f64 3.5

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot x - 1, x, \sqrt{x}\right)}{x} \]

5. Applied rewrites 3.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot x - 1, x, \sqrt{x}\right)}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\sqrt{x}}{x} \]
7. Step-by-step derivation

   1. lift-sqrt.f64 5.8

      \[\leadsto \frac{\sqrt{x}}{x} \]

8. Applied rewrites 5.8%

   \[\leadsto \frac{\sqrt{x}}{x} \]
9. Add Preprocessing

Alternative 10: 3.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 x))

C

double code(double x) {
	return 0.5 * 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 = 0.5d0 * x
end function

Java

public static double code(double x) {
	return 0.5 * x;
}

Python

def code(x):
	return 0.5 * x

Julia

function code(x)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * x;
end

Wolfram

code[x_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sqrt{x} + x \cdot \left(\frac{1}{2} \cdot x - 1\right)}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot x - 1, x, \sqrt{x}\right)}{x} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot x - 1, x, \sqrt{x}\right)}{x} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot x - 1, x, \sqrt{x}\right)}{x} \]

   7. lift-sqrt.f64 3.5

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot x - 1, x, \sqrt{x}\right)}{x} \]

5. Applied rewrites 3.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot x - 1, x, \sqrt{x}\right)}{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{x} \]
7. Step-by-step derivation

   1. lift-*.f64 3.6

      \[\leadsto 0.5 \cdot x \]

8. Applied rewrites 3.6%

   \[\leadsto 0.5 \cdot \color{blue}{x} \]
9. Add Preprocessing

Developer Target 1: 38.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} - {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (pow x -0.5) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return pow(x, -0.5) - pow((x + 1.0), -0.5);
}

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 = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025072 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))