2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.6%

Time: 4.1s

Alternatives: 9

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.6% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+52}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \sqrt{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+52)
   (/ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (sqrt (fma x x x)))
   (/ (/ (* 0.5 (sqrt x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+52) {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(x))) / sqrt(fma(x, x, x));
	} else {
		tmp = ((0.5 * sqrt(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+52)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) / sqrt(fma(x, x, x)));
	else
		tmp = Float64(Float64(Float64(0.5 * sqrt(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+52], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999999e51

   1. Initial program 21.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-+.f64 21.3

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

   3. Applied rewrites 21.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. flip-- N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-prod N/A

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

      6. *-commutative N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. rem-square-sqrt N/A

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

      11. pow2 N/A

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

      12. metadata-eval N/A

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

      13. unpow-prod-down N/A

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

      14. *-rgt-identity N/A

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

      15. pow2 N/A

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

      16. rem-square-sqrt N/A

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

      17. lower-fma.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 9.9999999999999999e51 < x

   1. Initial program 42.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 79.3

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

   7. Applied rewrites 79.3%

      \[\leadsto \frac{0.5 \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 99.7%

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

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{\frac{1}{t\_0 + \sqrt{x}}}{\sqrt{x} \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0)))) (/ (/ 1.0 (+ t_0 (sqrt x))) (* (sqrt x) t_0))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return (1.0 / (t_0 + sqrt(x))) / (sqrt(x) * t_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
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = (1.0d0 / (t_0 + sqrt(x))) / (sqrt(x) * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	return (1.0 / (t_0 + Math.sqrt(x))) / (Math.sqrt(x) * t_0);
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	return (1.0 / (t_0 + math.sqrt(x))) / (math.sqrt(x) * t_0)

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	return Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(sqrt(x) * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = (1.0 / (t_0 + sqrt(x))) / (sqrt(x) * t_0);
end

Wolfram

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

Derivation

1. Initial program 38.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-+.f64 38.7

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

3. Applied rewrites 38.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 39.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.2%

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

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, x\right)} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \sqrt{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+30)
   (/ 1.0 (* (sqrt (fma x x x)) (+ (sqrt (+ x 1.0)) (sqrt x))))
   (/ (/ (* 0.5 (sqrt x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+30) {
		tmp = 1.0 / (sqrt(fma(x, x, x)) * (sqrt((x + 1.0)) + sqrt(x)));
	} else {
		tmp = ((0.5 * sqrt(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+30)
		tmp = Float64(1.0 / Float64(sqrt(fma(x, x, x)) * Float64(sqrt(Float64(x + 1.0)) + sqrt(x))));
	else
		tmp = Float64(Float64(Float64(0.5 * sqrt(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+30], N[(1.0 / N[(N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e30

   1. Initial program 34.2%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. lift-+.f64 34.7

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

   3. Applied rewrites 34.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. flip-- N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. associate-/l/ N/A

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

   10. Applied rewrites 99.4%

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

   if 2e30 < x

   1. Initial program 39.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 80.9

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

   7. Applied rewrites 80.9%

      \[\leadsto \frac{0.5 \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 99.7%

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

Alternative 4: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-+.f64 38.7

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

3. Applied rewrites 38.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 39.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around -inf

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

    1. mul-1-neg N/A

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

    2. sqrt-pow2 N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

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

    5. *-commutative N/A

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

    6. mul-1-neg N/A

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

    7. lower-neg.f64 N/A

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

    8. lower-neg.f64 97.7

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

11. Applied rewrites 97.7%

    \[\leadsto \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\color{blue}{-\left(-x\right)}} \]
12. Add Preprocessing

Alternative 5: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((0.5 * sqrt(x)) / x) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = ((0.5d0 * sqrt(x)) / x) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 81.6%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 80.6

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

7. Applied rewrites 80.6%

   \[\leadsto \frac{0.5 \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 97.6%

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

Alternative 6: 97.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (0.5 / sqrt(x)) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (0.5d0 / sqrt(x)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out-- N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. inv-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 80.7

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

4. Applied rewrites 80.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left({\left(\sqrt{x}\right)}^{-1} - \sqrt{x}\right)}{x \cdot x}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 97.7%

   \[\leadsto \frac{\frac{\left({x}^{-0.5} - \sqrt{x}\right) \cdot -0.5}{x}}{\color{blue}{x}} \]

7. Taylor expanded in x around inf

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. lift-sqrt.f64 97.6

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

9. Applied rewrites 97.6%

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

Alternative 7: 80.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (0.5 * sqrt(x)) / (x * x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (0.5d0 * sqrt(x)) / (x * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 81.6%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 80.6

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

7. Applied rewrites 80.6%

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

Alternative 8: 36.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = ((x + 1.0d0) - x) / (sqrt(x) + x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 29.7

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

3. Applied rewrites 29.7%

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

   1. lift-pow.f64 N/A

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

   2. inv-pow N/A

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

   3. lower-/.f64 29.7

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

5. Applied rewrites 29.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. frac-sub N/A

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

   10. div-sub N/A

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

   11. *-lft-identity N/A

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

   12. *-rgt-identity N/A

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

   13. div-sub N/A

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

   14. flip-- N/A

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

7. Applied rewrites 40.7%

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

8. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. rem-square-sqrt N/A

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

   4. lower-+.f64 N/A

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

   5. lift-sqrt.f64 36.2

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

10. Applied rewrites 36.2%

    \[\leadsto \frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x} + x}} \]
11. Add Preprocessing

Alternative 9: 35.7% accurate, 49.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 38.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-+.f64 38.7

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

3. Applied rewrites 38.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. flip-- N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 39.5%

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

6. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. sqrt-pow2 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 35.7

       \[\leadsto \frac{0.5}{\sqrt{x}} \cdot 0 \]

8. Applied rewrites 35.7%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}} \cdot 0} \]

9. Taylor expanded in x around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 35.7%

    \[\leadsto 0 \]
12. Add Preprocessing

Developer Target 1: 38.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025092 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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