ENA, Section 1.4, Exercise 4d

Percentage Accurate: 60.8% → 99.5%

Time: 9.9s

Alternatives: 7

Speedup: 1.0×

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

real(8) function code(x, eps)
    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: 60.8% 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

real(8) function code(x, eps)
    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, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps / (x + pow(((x * x) - eps), 0.5));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (x + (((x * x) - eps) ** 0.5d0))
end function

Java

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

Python

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

Julia

function code(x, eps)
	return Float64(eps / Float64(x + (Float64(Float64(x * x) - eps) ^ 0.5)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (x + (((x * x) - eps) ^ 0.5));
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\left(x + \sqrt{x \cdot x - \varepsilon}\right)}\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(\color{blue}{x} + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x - \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   12. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(x \cdot x - \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   14. *-lowering-*.f64 62.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 62.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.5%

   \[\leadsto \frac{\color{blue}{\varepsilon}}{x + {\left(x \cdot x - \varepsilon\right)}^{0.5}} \]
8. Add Preprocessing

Alternative 2: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-148}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot 2 + \varepsilon \cdot \frac{-0.5 + -0.125 \cdot \frac{\varepsilon}{x \cdot x}}{x}}{\varepsilon}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-148)
     t_0
     (/
      1.0
      (/
       (+ (* x 2.0) (* eps (/ (+ -0.5 (* -0.125 (/ eps (* x x)))) x)))
       eps)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-148) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((x * 2.0) + (eps * ((-0.5 + (-0.125 * (eps / (x * x)))) / x))) / eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-4d-148)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((x * 2.0d0) + (eps * (((-0.5d0) + ((-0.125d0) * (eps / (x * x)))) / x))) / eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-148) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((x * 2.0) + (eps * ((-0.5 + (-0.125 * (eps / (x * x)))) / x))) / eps);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -4e-148:
		tmp = t_0
	else:
		tmp = 1.0 / (((x * 2.0) + (eps * ((-0.5 + (-0.125 * (eps / (x * x)))) / x))) / eps)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -4e-148)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * 2.0) + Float64(eps * Float64(Float64(-0.5 + Float64(-0.125 * Float64(eps / Float64(x * x)))) / x))) / eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -4e-148)
		tmp = t_0;
	else
		tmp = 1.0 / (((x * 2.0) + (eps * ((-0.5 + (-0.125 * (eps / (x * x)))) / x))) / eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-148], t$95$0, N[(1.0 / N[(N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(N[(-0.5 + N[(-0.125 * N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -3.99999999999999974e-148

   1. Initial program 98.8%

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

   if -3.99999999999999974e-148 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.3%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(\frac{x + \sqrt{x \cdot x - \varepsilon}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\color{blue}{x + \sqrt{x \cdot x - \varepsilon}}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{\frac{\color{blue}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}}{x + \sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

      8. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{x - \color{blue}{\sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(x - \sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\color{blue}{x} - \sqrt{x \cdot x - \varepsilon}\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   4. Applied egg-rr 8.3%

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

   5. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2 \cdot x + \varepsilon \cdot \left(\frac{-1}{8} \cdot \frac{\varepsilon}{{x}^{3}} - \frac{1}{2} \cdot \frac{1}{x}\right)}{\varepsilon}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot x + \varepsilon \cdot \left(\frac{-1}{8} \cdot \frac{\varepsilon}{{x}^{3}} - \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{\varepsilon}\right)\right) \]

   7. Simplified 98.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot 2 + \varepsilon \cdot \frac{-0.5 + -0.125 \cdot \frac{\varepsilon}{x \cdot x}}{x}}{\varepsilon}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-90}:\\ \;\;\;\;x - \sqrt{0 - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}{\varepsilon}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.1e-90)
   (- x (sqrt (- 0.0 eps)))
   (/ 1.0 (/ (+ (* x 2.0) (/ (* eps -0.5) x)) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.1e-90) {
		tmp = x - sqrt((0.0 - eps));
	} else {
		tmp = 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 3.1d-90) then
        tmp = x - sqrt((0.0d0 - eps))
    else
        tmp = 1.0d0 / (((x * 2.0d0) + ((eps * (-0.5d0)) / x)) / eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.1e-90) {
		tmp = x - Math.sqrt((0.0 - eps));
	} else {
		tmp = 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.1e-90:
		tmp = x - math.sqrt((0.0 - eps))
	else:
		tmp = 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.1e-90)
		tmp = Float64(x - sqrt(Float64(0.0 - eps)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * 2.0) + Float64(Float64(eps * -0.5) / x)) / eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.1e-90)
		tmp = x - sqrt((0.0 - eps));
	else
		tmp = 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.1e-90], N[(x - N[Sqrt[N[(0.0 - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1000000000000001e-90

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \varepsilon\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\left(0 - \varepsilon\right)\right)\right) \]

      3. --lowering--.f64 90.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \varepsilon\right)\right)\right) \]

   5. Simplified 90.5%

      \[\leadsto x - \sqrt{\color{blue}{0 - \varepsilon}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 90.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\varepsilon\right)\right)\right) \]

   7. Applied egg-rr 90.5%

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

   if 3.1000000000000001e-90 < x

   1. Initial program 23.3%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(\frac{x + \sqrt{x \cdot x - \varepsilon}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\color{blue}{x + \sqrt{x \cdot x - \varepsilon}}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{\frac{\color{blue}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}}{x + \sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

      8. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{x - \color{blue}{\sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(x - \sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\color{blue}{x} - \sqrt{x \cdot x - \varepsilon}\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   4. Applied egg-rr 23.2%

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

   5. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{\varepsilon}{x} + 2 \cdot x}{\varepsilon}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{\varepsilon}{x} + 2 \cdot x\right), \color{blue}{\varepsilon}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot x + \frac{-1}{2} \cdot \frac{\varepsilon}{x}\right), \varepsilon\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{-1}{2} \cdot \varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), x\right)\right), \varepsilon\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), x\right)\right), \varepsilon\right)\right) \]

      9. *-lowering-*.f64 83.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), x\right)\right), \varepsilon\right)\right) \]

   7. Simplified 83.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}{\varepsilon}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-90}:\\ \;\;\;\;x - \sqrt{0 - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}{\varepsilon}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ 1.0 (/ (+ (* x 2.0) (/ (* eps -0.5) x)) eps)))

C

double code(double x, double eps) {
	return 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 / (((x * 2.0d0) + ((eps * (-0.5d0)) / x)) / eps)
end function

Java

public static double code(double x, double eps) {
	return 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps);
}

Python

def code(x, eps):
	return 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps)

Julia

function code(x, eps)
	return Float64(1.0 / Float64(Float64(Float64(x * 2.0) + Float64(Float64(eps * -0.5) / x)) / eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 / (((x * 2.0) + ((eps * -0.5) / x)) / eps);
end

Wolfram

code[x_, eps_] := N[(1.0 / N[(N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(\frac{x + \sqrt{x \cdot x - \varepsilon}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\color{blue}{x + \sqrt{x \cdot x - \varepsilon}}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{\frac{\color{blue}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}}{x + \sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

   8. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{x - \color{blue}{\sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(x - \sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\color{blue}{x} - \sqrt{x \cdot x - \varepsilon}\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right)\right) \]

   12. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

   13. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

4. Applied egg-rr 62.2%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{\varepsilon}{x} + 2 \cdot x}{\varepsilon}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{\varepsilon}{x} + 2 \cdot x\right), \color{blue}{\varepsilon}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot x + \frac{-1}{2} \cdot \frac{\varepsilon}{x}\right), \varepsilon\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{-1}{2} \cdot \varepsilon}{x}\right)\right), \varepsilon\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), x\right)\right), \varepsilon\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), x\right)\right), \varepsilon\right)\right) \]

   9. *-lowering-*.f64 44.4%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), x\right)\right), \varepsilon\right)\right) \]

7. Simplified 44.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}{\varepsilon}}} \]
8. Add Preprocessing

Alternative 5: 45.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return (eps * 0.5) / x

Julia

function code(x, eps)
	return Float64(Float64(eps * 0.5) / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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

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

   2. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right), \color{blue}{x}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\varepsilon \cdot \frac{-1}{2}\right)\right), x\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\varepsilon \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), x\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right), x\right) \]

   8. *-lowering-*.f64 44.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right), x\right) \]

5. Simplified 44.0%

   \[\leadsto \color{blue}{\frac{\varepsilon \cdot 0.5}{x}} \]
6. Add Preprocessing

Alternative 6: 45.4% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * (0.5 / x)

Julia

function code(x, eps)
	return Float64(eps * Float64(0.5 / x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(\frac{x + \sqrt{x \cdot x - \varepsilon}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\color{blue}{x + \sqrt{x \cdot x - \varepsilon}}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{\frac{\color{blue}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}}{x + \sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

   8. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} + \frac{1}{2}}{x - \color{blue}{\sqrt{x \cdot x - \varepsilon}}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2}\right), \color{blue}{\left(x - \sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\color{blue}{x} - \sqrt{x \cdot x - \varepsilon}\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right)\right) \]

   12. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right)\right) \]

   13. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

4. Applied egg-rr 62.2%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x}\right)}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{\frac{1}{2}}{x}\right)\right) \]

   9. /-lowering-/.f64 43.8%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right) \]

7. Simplified 43.8%

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

Alternative 7: 4.3% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{x}\right) \]
4. Step-by-step derivation

5. Simplified 4.3%

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

   1. +-inverses 4.3%

      \[\leadsto 0 \]

7. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.5% accurate, 1.0× 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

real(8) function code(x, eps)
    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 2024138 
(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))))