ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.1% → 99.5%

Time: 5.2s

Alternatives: 7

Speedup: 0.7×

Specification

\[\left(0 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[\begin{array}{l} \\ x - \sqrt{x \cdot x - \varepsilon} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- x (sqrt (- (* x x) eps))))

C

double code(double x, double eps) {
	return x - sqrt(((x * x) - eps));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x - sqrt(((x * x) - eps))
end function

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}

Python

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))

Julia

function code(x, eps)
	return Float64(x - sqrt(Float64(Float64(x * x) - eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = x - sqrt(((x * x) - eps));
end

Wolfram

code[x_, eps_] := N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \sqrt{x \cdot x - \varepsilon} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- x (sqrt (- (* x x) eps))))

C

double code(double x, double eps) {
	return x - sqrt(((x * x) - eps));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x - sqrt(((x * x) - eps))
end function

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}

Python

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))

Julia

function code(x, eps)
	return Float64(x - sqrt(Float64(Float64(x * x) - eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = x - sqrt(((x * x) - eps));
end

Wolfram

code[x_, eps_] := N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps / (sqrt(((x * x) - eps)) + 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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (sqrt(((x * x) - eps)) + x)
end function

Java

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

Python

def code(x, eps):
	return eps / (math.sqrt(((x * x) - eps)) + x)

Julia

function code(x, eps)
	return Float64(eps / Float64(sqrt(Float64(Float64(x * x) - eps)) + x))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (sqrt(((x * x) - eps)) + x);
end

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. div-sub N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 62.6

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

4. Applied rewrites 62.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. lift-/.f64 N/A

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

   6. associate--r- N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{\varepsilon}{\sqrt{x \cdot x - \varepsilon} + x} \]
8. Add Preprocessing

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, 0.125, 0.5\right)}{x} \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (fma x x (- eps))))
   (* (/ (fma (/ eps (* x x)) 0.125 0.5) x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - sqrt(fma(x, x, -eps));
	} else {
		tmp = (fma((eps / (x * x)), 0.125, 0.5) / x) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-153)
		tmp = Float64(x - sqrt(fma(x, x, Float64(-eps))));
	else
		tmp = Float64(Float64(fma(Float64(eps / Float64(x * x)), 0.125, 0.5) / x) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-153], N[(x - N[Sqrt[N[(x * x + (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.125 + 0.5), $MachinePrecision] / x), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 98.7%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon}}{x} \]

      2. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}}{x} + \frac{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon}{x}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot {\varepsilon}^{2}}{{x}^{2}}}}{x} + \frac{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon}{x} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot {\varepsilon}^{2}}{{x}^{2} \cdot x}} + \frac{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot {\varepsilon}^{2}}{\color{blue}{\left(x \cdot x\right)} \cdot x} + \frac{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon}{x} \]

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}}\right) \cdot \varepsilon + \frac{\color{blue}{\frac{1}{2}} \cdot \varepsilon}{x} \]

      12. associate-*l/ N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}}\right) \cdot \varepsilon + \color{blue}{\frac{\frac{1}{2}}{x} \cdot \varepsilon} \]

      13. metadata-eval N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}}\right) \cdot \varepsilon + \frac{\color{blue}{\frac{1}{2} \cdot 1}}{x} \cdot \varepsilon \]

      14. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}}\right) \cdot \varepsilon + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x}\right)} \cdot \varepsilon \]

      15. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)} \]

      16. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon} \]

   5. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{{x}^{3}}, \varepsilon, \frac{0.5}{x}\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, 0.125, 0.5\right)}{x} \cdot \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, 0.125, 0.5\right)}{x} \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (fma x x (- eps))))
   (* (/ eps x) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - sqrt(fma(x, x, -eps));
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-153)
		tmp = Float64(x - sqrt(fma(x, x, Float64(-eps))));
	else
		tmp = Float64(Float64(eps / x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-153], N[(x - N[Sqrt[N[(x * x + (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 98.7%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.5

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

   8. Applied rewrites 96.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (- eps)))
   (* (/ eps x) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-5d-153)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (eps / x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-153:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (eps / x) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-153)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(eps / x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153)
		tmp = x - sqrt(-eps);
	else
		tmp = (eps / x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-153], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 98.7%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.0

         \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   5. Applied rewrites 95.0%

      \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 9.6

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.5

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

   8. Applied rewrites 96.5%

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

Alternative 5: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (- eps)))
   (* (/ 0.5 x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-5d-153)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (0.5d0 / x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-153:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (0.5 / x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-153)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(0.5 / x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153)
		tmp = x - sqrt(-eps);
	else
		tmp = (0.5 / x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-153], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 98.7%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.0

         \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   5. Applied rewrites 95.0%

      \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 6: 57.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x - \sqrt{-\varepsilon} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- x (sqrt (- eps))))

C

double code(double x, double eps) {
	return x - sqrt(-eps);
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x - sqrt(-eps)
end function

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(-eps);
}

Python

def code(x, eps):
	return x - math.sqrt(-eps)

Julia

function code(x, eps)
	return Float64(x - sqrt(Float64(-eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = x - sqrt(-eps);
end

Wolfram

code[x_, eps_] := N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 57.3

      \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

5. Applied rewrites 57.3%

   \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]
6. Add Preprocessing

Alternative 7: 3.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x - \left(-x\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- x (- x)))

C

double code(double x, double eps) {
	return 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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x - -x
end function

Java

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

Python

def code(x, eps):
	return x - -x

Julia

function code(x, eps)
	return Float64(x - Float64(-x))
end

MATLAB

function tmp = code(x, eps)
	tmp = x - -x;
end

Wolfram

code[x_, eps_] := N[(x - (-x)), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 3.6

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

5. Applied rewrites 3.6%

   \[\leadsto x - \color{blue}{\left(-x\right)} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps / (x + sqrt(((x * x) - eps)));
}

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

Java

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

Python

def code(x, eps):
	return eps / (x + math.sqrt(((x * x) - eps)))

Julia

function code(x, eps)
	return Float64(eps / Float64(x + sqrt(Float64(Float64(x * x) - eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (x + sqrt(((x * x) - eps)));
end

Wolfram

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

Reproduce

herbie shell --seed 2025017 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4d"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))

  :alt
  (! :herbie-platform default (/ eps (+ x (sqrt (- (* x x) eps)))))

  (- x (sqrt (- (* x x) eps))))