2isqrt (example 3.6)

Percentage Accurate: 40.1% → 99.9%

Time: 7.2s

Alternatives: 10

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.1% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x - -1}\\ \mathbf{if}\;x \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{t\_0 \cdot \mathsf{fma}\left(\sqrt{x}, t\_0, x\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- x -1.0))))
   (if (<= x 4e+14)
     (/ (- (- x -1.0) x) (* t_0 (fma (sqrt x) t_0 x)))
     (* (pow x -1.5) 0.5))))

C

double code(double x) {
	double t_0 = sqrt((x - -1.0));
	double tmp;
	if (x <= 4e+14) {
		tmp = ((x - -1.0) - x) / (t_0 * fma(sqrt(x), t_0, x));
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 4e+14)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64(t_0 * fma(sqrt(x), t_0, x)));
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4e+14], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(t$95$0 * N[(N[Sqrt[x], $MachinePrecision] * t$95$0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4e14

   1. Initial program 61.2%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lower-/.f64 N/A

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

      6. *-lft-identity N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 63.4

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

      20. lift-+.f64 N/A

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

      21. metadata-eval N/A

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

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

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-prod N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-rgt-out N/A

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

   8. Applied rewrites 99.6%

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

   if 4e14 < x

   1. Initial program 37.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{-0.5 \cdot \left(\frac{0.75 \cdot t\_0}{x} - t\_0\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x)))) (/ (* -0.5 (- (/ (* 0.75 t_0) x) t_0)) x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return (-0.5 * (((0.75 * t_0) / x) - t_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
    real(8) :: t_0
    t_0 = sqrt((1.0d0 / x))
    code = ((-0.5d0) * (((0.75d0 * t_0) / x) - t_0)) / x
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	return (-0.5 * (((0.75 * t_0) / x) - t_0)) / x;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	return (-0.5 * (((0.75 * t_0) / x) - t_0)) / x

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(-0.5 * Float64(Float64(Float64(0.75 * t_0) / x) - t_0)) / x)
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / x));
	tmp = (-0.5 * (((0.75 * t_0) / x) - t_0)) / x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(-0.5 * N[(N[(N[(0.75 * t$95$0), $MachinePrecision] / x), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 39.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 86.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 98.3%

   \[\leadsto \frac{-0.5 \cdot \left(\frac{0.75 \cdot \sqrt{\frac{1}{x}}}{x} - \sqrt{\frac{1}{x}}\right)}{\color{blue}{x}} \]
9. Add Preprocessing

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, 0.5 \cdot \sqrt{x}\right)}{x}}{0.5 + x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (fma -0.125 (sqrt (/ 1.0 x)) (* 0.5 (sqrt x))) x) (+ 0.5 x)))

C

double code(double x) {
	return (fma(-0.125, sqrt((1.0 / x)), (0.5 * sqrt(x))) / x) / (0.5 + x);
}

Julia

function code(x)
	return Float64(Float64(fma(-0.125, sqrt(Float64(1.0 / x)), Float64(0.5 * sqrt(x))) / x) / Float64(0.5 + x))
end

Wolfram

code[x_] := N[(N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(0.5 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

   6. *-lft-identity N/A

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

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. sqrt-unprod N/A

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

   18. lower-sqrt.f64 N/A

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

   19. lower-*.f64 39.4

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

   20. lift-+.f64 N/A

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

   21. metadata-eval N/A

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

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

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

4. Applied rewrites 39.4%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 38.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 98.3%

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, 0.5 \cdot \sqrt{x}\right)}{x}}}{0.5 + x} \]
11. Add Preprocessing

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{\left(\sqrt{x - -1} + \sqrt{x}\right) \cdot \sqrt{\left(x - -1\right) \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+137)
   (/ 1.0 (* (+ (sqrt (- x -1.0)) (sqrt x)) (sqrt (* (- x -1.0) x))))
   (/ (* (sqrt (/ 1.0 x)) 0.5) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+137) {
		tmp = 1.0 / ((sqrt((x - -1.0)) + sqrt(x)) * sqrt(((x - -1.0) * x)));
	} else {
		tmp = (sqrt((1.0 / x)) * 0.5) / 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 <= 2d+137) then
        tmp = 1.0d0 / ((sqrt((x - (-1.0d0))) + sqrt(x)) * sqrt(((x - (-1.0d0)) * x)))
    else
        tmp = (sqrt((1.0d0 / x)) * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+137) {
		tmp = 1.0 / ((Math.sqrt((x - -1.0)) + Math.sqrt(x)) * Math.sqrt(((x - -1.0) * x)));
	} else {
		tmp = (Math.sqrt((1.0 / x)) * 0.5) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e+137:
		tmp = 1.0 / ((math.sqrt((x - -1.0)) + math.sqrt(x)) * math.sqrt(((x - -1.0) * x)))
	else:
		tmp = (math.sqrt((1.0 / x)) * 0.5) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+137)
		tmp = Float64(1.0 / Float64(Float64(sqrt(Float64(x - -1.0)) + sqrt(x)) * sqrt(Float64(Float64(x - -1.0) * x))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e+137)
		tmp = 1.0 / ((sqrt((x - -1.0)) + sqrt(x)) * sqrt(((x - -1.0) * x)));
	else
		tmp = (sqrt((1.0 / x)) * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e+137], N[(1.0 / 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]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e137

   1. Initial program 11.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lower-/.f64 N/A

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

      6. *-lft-identity N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 12.2

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

      20. lift-+.f64 N/A

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

      21. metadata-eval N/A

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

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

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 12.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 17.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.5%

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

   if 2.0000000000000001e137 < x

   1. Initial program 69.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

      \[\leadsto \frac{-0.5 \cdot \left(\frac{0.75 \cdot \sqrt{\frac{1}{x}}}{x} - \sqrt{\frac{1}{x}}\right)}{\color{blue}{x}} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.9%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / ((sqrt(x) + sqrt((x - -1.0))) * (0.5 + x));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

   6. *-lft-identity N/A

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

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. sqrt-unprod N/A

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

   18. lower-sqrt.f64 N/A

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

   19. lower-*.f64 39.4

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

   20. lift-+.f64 N/A

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

   21. metadata-eval N/A

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

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

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

4. Applied rewrites 39.4%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 38.4%

   \[\leadsto \frac{\sqrt{x - -1} - \sqrt{x}}{\color{blue}{0.5 + x}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lift--.f64 N/A

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

   13. lower-/.f64 N/A

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

9. Applied rewrites 40.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 97.4%

    \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{x} + \sqrt{x - -1}\right) \cdot \left(0.5 + x\right)} \]
13. Add Preprocessing

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (sqrt((1.0 / x)) * 0.5) / (0.5 + x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

   6. *-lft-identity N/A

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

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. sqrt-unprod N/A

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

   18. lower-sqrt.f64 N/A

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

   19. lower-*.f64 39.4

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

   20. lift-+.f64 N/A

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

   21. metadata-eval N/A

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

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

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

4. Applied rewrites 39.4%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 38.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 96.9%

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

Alternative 7: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (sqrt((1.0 / x)) * 0.5) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 86.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 98.3%

   \[\leadsto \frac{-0.5 \cdot \left(\frac{0.75 \cdot \sqrt{\frac{1}{x}}}{x} - \sqrt{\frac{1}{x}}\right)}{\color{blue}{x}} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 96.7%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} \]
12. Add Preprocessing

Alternative 8: 81.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 86.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 85.0%

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

Alternative 9: 37.8% 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 39.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

   6. *-lft-identity N/A

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

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. sqrt-unprod N/A

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

   18. lower-sqrt.f64 N/A

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

   19. lower-*.f64 39.4

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

   20. lift-+.f64 N/A

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

   21. metadata-eval N/A

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

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

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

4. Applied rewrites 39.4%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 41.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 35.8%

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

Alternative 10: 5.6% accurate, 2.2× 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 39.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.8%

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

Developer Target 1: 40.1% 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 2025019 
(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)))))