2isqrt (example 3.6)

Percentage Accurate: 38.0% → 98.7%

Time: 3.7s

Alternatives: 12

Speedup: 1.8×

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: 38.0% 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: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+15}:\\ \;\;\;\;\frac{\sqrt{\left(x - -1\right) \cdot x} - x \cdot 1}{x \cdot \sqrt{x - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+15)
   (/ (- (sqrt (* (- x -1.0) x)) (* x 1.0)) (* x (sqrt (- x -1.0))))
   (/ (/ 0.5 x) (* (sqrt (/ 1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+15) {
		tmp = (sqrt(((x - -1.0) * x)) - (x * 1.0)) / (x * sqrt((x - -1.0)));
	} else {
		tmp = (0.5 / x) / (sqrt((1.0 / 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 <= 1d+15) then
        tmp = (sqrt(((x - (-1.0d0)) * x)) - (x * 1.0d0)) / (x * sqrt((x - (-1.0d0))))
    else
        tmp = (0.5d0 / x) / (sqrt((1.0d0 / x)) * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+15) {
		tmp = (Math.sqrt(((x - -1.0) * x)) - (x * 1.0)) / (x * Math.sqrt((x - -1.0)));
	} else {
		tmp = (0.5 / x) / (Math.sqrt((1.0 / x)) * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+15:
		tmp = (math.sqrt(((x - -1.0) * x)) - (x * 1.0)) / (x * math.sqrt((x - -1.0)))
	else:
		tmp = (0.5 / x) / (math.sqrt((1.0 / x)) * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+15)
		tmp = Float64(Float64(sqrt(Float64(Float64(x - -1.0) * x)) - Float64(x * 1.0)) / Float64(x * sqrt(Float64(x - -1.0))));
	else
		tmp = Float64(Float64(0.5 / x) / Float64(sqrt(Float64(1.0 / x)) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+15)
		tmp = (sqrt(((x - -1.0) * x)) - (x * 1.0)) / (x * sqrt((x - -1.0)));
	else
		tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+15], N[(N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - N[(x * 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e15

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. 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. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div-sound-left N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. pow-flip N/A

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

      12. pow1/2 N/A

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

      13. mult-flip N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 21.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 6.5%

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

   if 1e15 < x

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. rem-square-sqrt N/A

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

      3. associate-*r* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. fabs-pow2-rev N/A

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

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

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

      12. sqrt-unprod N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

      13. inv-pow N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

      15. metadata-eval N/A

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

      16. lower-sqrt.f64 N/A

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

      17. unpow3 N/A

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. fabs-pow2-rev N/A

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

      7. sqrt-prod-sound N/A

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

      8. fabs-pow2-rev N/A

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

      9. pow2 N/A

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

      10. sqrt-unprod N/A

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

      11. rem-square-sqrt N/A

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

      12. associate-/r* N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 97.6

         \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 86000000:\\ \;\;\;\;\frac{\sqrt{x - -1} - \sqrt{x}}{\sqrt{\left(x - -1\right) \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 86000000.0)
   (/ (- (sqrt (- x -1.0)) (sqrt x)) (sqrt (* (- x -1.0) x)))
   (/ (/ 0.5 x) (* (sqrt (/ 1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 86000000.0) {
		tmp = (sqrt((x - -1.0)) - sqrt(x)) / sqrt(((x - -1.0) * x));
	} else {
		tmp = (0.5 / x) / (sqrt((1.0 / 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 <= 86000000.0d0) then
        tmp = (sqrt((x - (-1.0d0))) - sqrt(x)) / sqrt(((x - (-1.0d0)) * x))
    else
        tmp = (0.5d0 / x) / (sqrt((1.0d0 / x)) * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 86000000.0) {
		tmp = (Math.sqrt((x - -1.0)) - Math.sqrt(x)) / Math.sqrt(((x - -1.0) * x));
	} else {
		tmp = (0.5 / x) / (Math.sqrt((1.0 / x)) * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 86000000.0:
		tmp = (math.sqrt((x - -1.0)) - math.sqrt(x)) / math.sqrt(((x - -1.0) * x))
	else:
		tmp = (0.5 / x) / (math.sqrt((1.0 / x)) * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 86000000.0)
		tmp = Float64(Float64(sqrt(Float64(x - -1.0)) - sqrt(x)) / sqrt(Float64(Float64(x - -1.0) * x)));
	else
		tmp = Float64(Float64(0.5 / x) / Float64(sqrt(Float64(1.0 / x)) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 86000000.0)
		tmp = (sqrt((x - -1.0)) - sqrt(x)) / sqrt(((x - -1.0) * x));
	else
		tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 86000000.0], N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.6e7

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. 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. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. *-commutative N/A

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

      18. sqrt-unprod N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. add-flip N/A

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

      22. metadata-eval N/A

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

      23. lower--.f64 38.0

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

   3. Applied rewrites 38.0%

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

   if 8.6e7 < x

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. rem-square-sqrt N/A

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

      3. associate-*r* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. fabs-pow2-rev N/A

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

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

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

      12. sqrt-unprod N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

      13. inv-pow N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

      15. metadata-eval N/A

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

      16. lower-sqrt.f64 N/A

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

      17. unpow3 N/A

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. fabs-pow2-rev N/A

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

      7. sqrt-prod-sound N/A

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

      8. fabs-pow2-rev N/A

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

      9. pow2 N/A

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

      10. sqrt-unprod N/A

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

      11. rem-square-sqrt N/A

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

      12. associate-/r* N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 97.6

         \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{x - -1}{x}} - 1}{\sqrt{x - -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))) 2e-14)
   (/ (/ 0.5 x) (* (sqrt (/ 1.0 x)) x))
   (/ (- (sqrt (/ (- x -1.0) x)) 1.0) (sqrt (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)))) <= 2e-14) {
		tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
	} else {
		tmp = (sqrt(((x - -1.0) / x)) - 1.0) / sqrt((x - -1.0));
	}
	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 (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))) <= 2d-14) then
        tmp = (0.5d0 / x) / (sqrt((1.0d0 / x)) * x)
    else
        tmp = (sqrt(((x - (-1.0d0)) / x)) - 1.0d0) / sqrt((x - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)))) <= 2e-14) {
		tmp = (0.5 / x) / (Math.sqrt((1.0 / x)) * x);
	} else {
		tmp = (Math.sqrt(((x - -1.0) / x)) - 1.0) / Math.sqrt((x - -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))) <= 2e-14:
		tmp = (0.5 / x) / (math.sqrt((1.0 / x)) * x)
	else:
		tmp = (math.sqrt(((x - -1.0) / x)) - 1.0) / math.sqrt((x - -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0)))) <= 2e-14)
		tmp = Float64(Float64(0.5 / x) / Float64(sqrt(Float64(1.0 / x)) * x));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x - -1.0) / x)) - 1.0) / sqrt(Float64(x - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)))) <= 2e-14)
		tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
	else
		tmp = (sqrt(((x - -1.0) / x)) - 1.0) / sqrt((x - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-14], N[(N[(0.5 / x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 2e-14

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. rem-square-sqrt N/A

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

      3. associate-*r* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. fabs-pow2-rev N/A

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

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

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

      12. sqrt-unprod N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

      13. inv-pow N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

      15. metadata-eval N/A

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

      16. lower-sqrt.f64 N/A

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

      17. unpow3 N/A

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. fabs-pow2-rev N/A

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

      7. sqrt-prod-sound N/A

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

      8. fabs-pow2-rev N/A

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

      9. pow2 N/A

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

      10. sqrt-unprod N/A

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

      11. rem-square-sqrt N/A

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

      12. associate-/r* N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 97.6

         \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{x}} \]

   if 2e-14 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. 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. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. *-commutative N/A

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

      18. sqrt-unprod N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. add-flip N/A

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

      22. metadata-eval N/A

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

      23. lower--.f64 38.0

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

   3. Applied rewrites 38.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. sqrt-unprod N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 38.0

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

   5. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 38.0

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

   9. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{x - -1}{x}} - 1}{\sqrt{x - -1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ t_1 := \frac{1}{\sqrt{x + 1}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - t\_1 \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{t\_0 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))) (t_1 (/ 1.0 (sqrt (+ x 1.0)))))
   (if (<= (- (/ 1.0 (sqrt x)) t_1) 1e-12)
     (/ (/ 0.5 x) (* t_0 x))
     (- t_0 t_1))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	double t_1 = 1.0 / sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - t_1) <= 1e-12) {
		tmp = (0.5 / x) / (t_0 * x);
	} else {
		tmp = t_0 - t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / x))
    t_1 = 1.0d0 / sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) - t_1) <= 1d-12) then
        tmp = (0.5d0 / x) / (t_0 * x)
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	double t_1 = 1.0 / Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - t_1) <= 1e-12) {
		tmp = (0.5 / x) / (t_0 * x);
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	t_1 = 1.0 / math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - t_1) <= 1e-12:
		tmp = (0.5 / x) / (t_0 * x)
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	t_1 = Float64(1.0 / sqrt(Float64(x + 1.0)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - t_1) <= 1e-12)
		tmp = Float64(Float64(0.5 / x) / Float64(t_0 * x));
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / x));
	t_1 = 1.0 / sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - t_1) <= 1e-12)
		tmp = (0.5 / x) / (t_0 * x);
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], 1e-12], N[(N[(0.5 / x), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 9.9999999999999998e-13

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. rem-square-sqrt N/A

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

      3. associate-*r* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. fabs-pow2-rev N/A

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

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

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

      12. sqrt-unprod N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

      13. inv-pow N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

      15. metadata-eval N/A

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

      16. lower-sqrt.f64 N/A

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

      17. unpow3 N/A

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. fabs-pow2-rev N/A

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

      7. sqrt-prod-sound N/A

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

      8. fabs-pow2-rev N/A

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

      9. pow2 N/A

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

      10. sqrt-unprod N/A

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

      11. rem-square-sqrt N/A

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

      12. associate-/r* N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 97.6

         \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{x}} \]

   if 9.9999999999999998e-13 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. 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. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div-sound-left N/A

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

      6. metadata-eval N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 29.5

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

   3. Applied rewrites 29.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} - \frac{1}{\sqrt{x + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x} - \frac{1}{\sqrt{x - -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))) 1e-12)
   (/ (/ 0.5 x) (* (sqrt (/ 1.0 x)) x))
   (- (/ (sqrt x) x) (/ 1.0 (sqrt (- x -1.0))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)))) <= 1e-12) {
		tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
	} else {
		tmp = (sqrt(x) / x) - (1.0 / sqrt((x - -1.0)));
	}
	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 (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))) <= 1d-12) then
        tmp = (0.5d0 / x) / (sqrt((1.0d0 / x)) * x)
    else
        tmp = (sqrt(x) / x) - (1.0d0 / sqrt((x - (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)))) <= 1e-12) {
		tmp = (0.5 / x) / (Math.sqrt((1.0 / x)) * x);
	} else {
		tmp = (Math.sqrt(x) / x) - (1.0 / Math.sqrt((x - -1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))) <= 1e-12:
		tmp = (0.5 / x) / (math.sqrt((1.0 / x)) * x)
	else:
		tmp = (math.sqrt(x) / x) - (1.0 / math.sqrt((x - -1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0)))) <= 1e-12)
		tmp = Float64(Float64(0.5 / x) / Float64(sqrt(Float64(1.0 / x)) * x));
	else
		tmp = Float64(Float64(sqrt(x) / x) - Float64(1.0 / sqrt(Float64(x - -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)))) <= 1e-12)
		tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
	else
		tmp = (sqrt(x) / x) - (1.0 / sqrt((x - -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[(0.5 / x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 9.9999999999999998e-13

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. rem-square-sqrt N/A

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

      3. associate-*r* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. fabs-pow2-rev N/A

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

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

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

      12. sqrt-unprod N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

      13. inv-pow N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

      15. metadata-eval N/A

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

      16. lower-sqrt.f64 N/A

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

      17. unpow3 N/A

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. fabs-pow2-rev N/A

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

      7. sqrt-prod-sound N/A

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

      8. fabs-pow2-rev N/A

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

      9. pow2 N/A

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

      10. sqrt-unprod N/A

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

      11. rem-square-sqrt N/A

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

      12. associate-/r* N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 97.6

         \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{x}} \]

   if 9.9999999999999998e-13 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. 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. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div-sound-left N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. pow-flip N/A

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

      12. pow1/2 N/A

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

      13. mult-flip N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 21.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. remove-double-div N/A

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

      5. lift-sqrt.f64 23.3

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

         \[\leadsto \frac{\sqrt{x}}{x} - \frac{1}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}} \]

      8. metadata-eval N/A

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

      9. lift--.f64 23.3

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

   5. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{x} - \frac{1}{\sqrt{x - -1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 72000000:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 72000000.0)
   (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (- x -1.0))))
   (/ (/ 0.5 x) (* (sqrt (/ 1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 72000000.0) {
		tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x - -1.0)));
	} else {
		tmp = (0.5 / x) / (sqrt((1.0 / 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 <= 72000000.0d0) then
        tmp = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x - (-1.0d0))))
    else
        tmp = (0.5d0 / x) / (sqrt((1.0d0 / x)) * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 72000000.0) {
		tmp = (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x - -1.0)));
	} else {
		tmp = (0.5 / x) / (Math.sqrt((1.0 / x)) * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 72000000.0:
		tmp = (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x - -1.0)))
	else:
		tmp = (0.5 / x) / (math.sqrt((1.0 / x)) * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 72000000.0)
		tmp = Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x - -1.0))));
	else
		tmp = Float64(Float64(0.5 / x) / Float64(sqrt(Float64(1.0 / x)) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 72000000.0)
		tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x - -1.0)));
	else
		tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 72000000.0], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e7

   1. Initial program 38.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

         \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}}} \]

      3. metadata-eval N/A

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

      4. lower--.f64 38.0

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

   3. Applied rewrites 38.0%

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

   if 7.2e7 < x

   1. Initial program 38.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. rem-square-sqrt N/A

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

      3. associate-*r* N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. fabs-pow2-rev N/A

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

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

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

      12. sqrt-unprod N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

      13. inv-pow N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

      15. metadata-eval N/A

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

      16. lower-sqrt.f64 N/A

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

      17. unpow3 N/A

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. fabs-pow2-rev N/A

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

      7. sqrt-prod-sound N/A

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

      8. fabs-pow2-rev N/A

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

      9. pow2 N/A

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

      10. sqrt-unprod N/A

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

      11. rem-square-sqrt N/A

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

      12. associate-/r* N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 97.6

         \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \]

   9. Applied rewrites 97.6%

      \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 x) (* (sqrt (/ 1.0 x)) x)))

C

double code(double x) {
	return (0.5 / x) / (sqrt((1.0 / 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 / x) / (sqrt((1.0d0 / x)) * x)
end function

Java

public static double code(double x) {
	return (0.5 / x) / (Math.sqrt((1.0 / x)) * x);
}

Python

def code(x):
	return (0.5 / x) / (math.sqrt((1.0 / x)) * x)

Julia

function code(x)
	return Float64(Float64(0.5 / x) / Float64(sqrt(Float64(1.0 / x)) * x))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / x) / (sqrt((1.0 / x)) * x);
end

Wolfram

code[x_] := N[(N[(0.5 / x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. rem-square-sqrt N/A

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

   3. associate-*r* N/A

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

   4. inv-pow N/A

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

   5. pow-prod-up N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. fabs-pow2-rev N/A

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

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

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

   10. pow-prod-up N/A

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

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

   12. sqrt-unprod N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

   13. inv-pow N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

   14. pow-prod-up N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

   15. metadata-eval N/A

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

   16. lower-sqrt.f64 N/A

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

   17. unpow3 N/A

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

4. Applied rewrites 64.5%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. fabs-pow2-rev N/A

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

   7. sqrt-prod-sound N/A

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

   8. fabs-pow2-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-unprod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-/r* N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 N/A

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

   15. mult-flip-rev N/A

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

   16. lower-/.f64 N/A

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

   17. lift-sqrt.f64 97.6

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 97.6

      \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot x} \]

9. Applied rewrites 97.6%

   \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{x}} \]
10. Add Preprocessing

Alternative 8: 97.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. rem-square-sqrt N/A

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

   3. associate-*r* N/A

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

   4. inv-pow N/A

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

   5. pow-prod-up N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. fabs-pow2-rev N/A

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

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

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

   10. pow-prod-up N/A

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

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

   12. sqrt-unprod N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

   13. inv-pow N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

   14. pow-prod-up N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

   15. metadata-eval N/A

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

   16. lower-sqrt.f64 N/A

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

   17. unpow3 N/A

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

4. Applied rewrites 64.5%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. fabs-pow2-rev N/A

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

   7. sqrt-prod-sound N/A

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

   8. fabs-pow2-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-unprod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-/r* N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 N/A

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

   15. mult-flip-rev N/A

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

   16. lower-/.f64 N/A

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

   17. lift-sqrt.f64 97.6

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

6. Applied rewrites 97.6%

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

Alternative 9: 96.3% accurate, 1.8× speedup

Math

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

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(0.5 / Float64(sqrt(x) * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. rem-square-sqrt N/A

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

   3. associate-*r* N/A

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

   4. inv-pow N/A

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

   5. pow-prod-up N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. fabs-pow2-rev N/A

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

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

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

   10. pow-prod-up N/A

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

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4}} \cdot \sqrt{\frac{1}{x}}} \]

   12. sqrt-unprod N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot \frac{1}{x}}} \]

   13. inv-pow N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{4} \cdot {x}^{-1}}} \]

   14. pow-prod-up N/A

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{{x}^{\left(4 + -1\right)}}} \]

   15. metadata-eval N/A

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

   16. lower-sqrt.f64 N/A

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

   17. unpow3 N/A

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

4. Applied rewrites 64.5%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}}} \]
5. Step-by-step derivation

   1. pow1/2 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   2. pow-flip 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   3. metadata-eval 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   4. metadata-eval 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   5. pow-prod-up 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   6. pow1/2 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   7. inv-pow 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   8. mult-flip 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   9. remove-double-div 64.5

      \[\leadsto \frac{0.5}{\sqrt{\left(x \cdot x\right) \cdot x}} \]

   10. lift-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. fabs-pow2-rev N/A

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

   15. sqrt-prod-sound N/A

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

   16. fabs-pow2-rev N/A

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

   17. pow2 N/A

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

   18. sqrt-unprod N/A

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

   19. rem-square-sqrt N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 N/A

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

   22. lift-sqrt.f64 96.3

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

6. Applied rewrites 96.3%

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

Alternative 10: 5.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	return sqrt((1.0 / 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((1.0d0 / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. 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. metadata-eval N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-div-sound-left N/A

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

   6. metadata-eval N/A

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

   7. pow1/2 N/A

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

   8. metadata-eval N/A

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

   9. pow-plus N/A

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

   10. metadata-eval N/A

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

   11. pow-flip N/A

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

   12. pow1/2 N/A

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

   13. mult-flip N/A

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

   14. lower-/.f64 N/A

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

3. Applied rewrites 21.1%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lift-sqrt.f64 5.6

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

6. Applied rewrites 5.6%

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

7. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

9. Applied rewrites 5.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
10. Add Preprocessing

Alternative 11: 3.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x, -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma 0.5 x -1.0))

C

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

Julia

function code(x)
	return fma(0.5, x, -1.0)
end

Wolfram

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

Derivation

1. Initial program 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. 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. sub-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\sqrt{1}}}{\sqrt{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left|1\right|}}}{\sqrt{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   10. sqrt-div-sound-left N/A

       \[\leadsto \color{blue}{\sqrt{\frac{\left|1\right|}{x}}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\color{blue}{1}}{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   12. rem-square-sqrt N/A

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

   13. sqrt-fabs-rev N/A

       \[\leadsto \sqrt{\color{blue}{\left|\sqrt{\frac{1}{x}}\right|} \cdot \sqrt{\frac{1}{x}}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   14. sqrt-prod-sound N/A

       \[\leadsto \color{blue}{\sqrt{\left|\sqrt{\frac{1}{x}}\right|} \cdot \sqrt{\sqrt{\frac{1}{x}}}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   15. sqrt-fabs-rev N/A

       \[\leadsto \sqrt{\left|\sqrt{\frac{1}{x}}\right|} \cdot \sqrt{\color{blue}{\left|\sqrt{\frac{1}{x}}\right|}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   16. lower-fma.f64 N/A

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

3. Applied rewrites 7.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. pow1/2 N/A

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

   13. pow-flip N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. pow-div N/A

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

   18. sqrt-pow2 N/A

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

   19. mult-flip N/A

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

5. Applied rewrites 5.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, {x}^{-2.5}, \frac{-1}{\sqrt{x - -1}}\right)} \]

6. Taylor expanded in x around 0

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

   1. sub-flip N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 3.4

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

8. Applied rewrites 3.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, -1\right)} \]
9. Add Preprocessing

Alternative 12: 2.5% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -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
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. 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. sub-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\sqrt{1}}}{\sqrt{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\left|1\right|}}}{\sqrt{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   10. sqrt-div-sound-left N/A

       \[\leadsto \color{blue}{\sqrt{\frac{\left|1\right|}{x}}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\color{blue}{1}}{x}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   12. rem-square-sqrt N/A

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

   13. sqrt-fabs-rev N/A

       \[\leadsto \sqrt{\color{blue}{\left|\sqrt{\frac{1}{x}}\right|} \cdot \sqrt{\frac{1}{x}}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   14. sqrt-prod-sound N/A

       \[\leadsto \color{blue}{\sqrt{\left|\sqrt{\frac{1}{x}}\right|} \cdot \sqrt{\sqrt{\frac{1}{x}}}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   15. sqrt-fabs-rev N/A

       \[\leadsto \sqrt{\left|\sqrt{\frac{1}{x}}\right|} \cdot \sqrt{\color{blue}{\left|\sqrt{\frac{1}{x}}\right|}} + \left(\mathsf{neg}\left(\frac{1}{\sqrt{x + 1}}\right)\right) \]

   16. lower-fma.f64 N/A

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

3. Applied rewrites 7.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. pow1/2 N/A

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

   13. pow-flip N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. pow-div N/A

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

   18. sqrt-pow2 N/A

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

   19. mult-flip N/A

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

5. Applied rewrites 5.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, {x}^{-2.5}, \frac{-1}{\sqrt{x - -1}}\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 2.5%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Developer Target 1: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

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

C

double code(double x) {
	return 1.0 / (((x + 1.0) * sqrt(x)) + (x * 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 / (((x + 1.0d0) * sqrt(x)) + (x * sqrt((x + 1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 38.0% accurate, 0.4× 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 2025136 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform c (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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

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