ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.4% → 98.4%

Time: 7.9s

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: 62.4% 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: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-151)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (+ (* x 2.0) (* -0.5 (/ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-151) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-151) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-151:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-151)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(-0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-151)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-151], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-151

   1. Initial program 99.3%

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

      1. flip-- 99.2%

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

      2. add-sqr-sqrt 98.6%

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

      3. div-sub 98.6%

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

      4. pow2 98.6%

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

      5. sub-neg 98.6%

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

      6. add-sqr-sqrt 98.6%

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

      7. hypot-define 98.6%

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

      8. pow2 98.6%

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

      9. sub-neg 98.6%

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

      10. add-sqr-sqrt 98.6%

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

      11. hypot-define 98.6%

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

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} - \frac{{x}^{2} - \varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
   5. Step-by-step derivation

      1. div-sub 98.6%

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

      2. sub-neg 98.6%

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

      3. associate--r+ 99.3%

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

      4. +-inverses 99.3%

         \[\leadsto \frac{\color{blue}{0} - \left(-\varepsilon\right)}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

      5. neg-sub0 99.3%

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

      6. remove-double-neg 99.3%

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

   6. Simplified 99.3%

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

   if -1.9999999999999999e-151 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.7%

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

      1. flip-- 6.8%

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

      2. add-sqr-sqrt 6.9%

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

      3. div-sub 6.8%

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

      4. pow2 6.8%

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

      5. sub-neg 6.8%

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

      6. add-sqr-sqrt 2.1%

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

      7. hypot-define 2.1%

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

      8. pow2 2.1%

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

      9. sub-neg 2.1%

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

      10. add-sqr-sqrt 2.1%

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

      11. hypot-define 2.1%

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

   4. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} - \frac{{x}^{2} - \varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
   5. Step-by-step derivation

      1. div-sub 2.1%

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

      2. sub-neg 2.1%

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

      3. associate--r+ 45.2%

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

      4. +-inverses 45.2%

         \[\leadsto \frac{\color{blue}{0} - \left(-\varepsilon\right)}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

      5. neg-sub0 45.2%

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

      6. remove-double-neg 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in eps around 0 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

      3. fma-define 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]

      4. associate-*r/ 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \left(\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}}\right)} \]

      5. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon\right)}}{x}\right)} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \varepsilon\right)}{x}\right)} \]

      7. rem-square-sqrt 99.8%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{-1} \cdot \varepsilon\right)}{x}\right)} \]

      8. neg-mul-1 99.8%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left(-\varepsilon\right)}}{x}\right)} \]

      9. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\color{blue}{-0.5 \cdot \varepsilon}}{x}\right)} \]

      10. distribute-frac-neg 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-\frac{0.5 \cdot \varepsilon}{x}}\right)} \]

      11. distribute-frac-neg2 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \varepsilon}{-x}}\right)} \]

      12. neg-mul-1 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \varepsilon}{\color{blue}{-1 \cdot x}}\right)} \]

      13. times-frac 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{-1} \cdot \frac{\varepsilon}{x}}\right)} \]

      14. metadata-eval 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-0.5} \cdot \frac{\varepsilon}{x}\right)} \]

   9. Simplified 99.8%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, -0.5 \cdot \frac{\varepsilon}{x}\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 99.8%

          \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}} \]

   11. Applied egg-rr 99.8%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-151) t_0 (/ eps (+ (* x 2.0) (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-151) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	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 <= (-2d-151)) then
        tmp = t_0
    else
        tmp = eps / ((x * 2.0d0) + ((-0.5d0) * (eps / x)))
    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 <= -2e-151) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-151:
		tmp = t_0
	else:
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -2e-151)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(-0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -2e-151)
		tmp = t_0;
	else
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	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, -2e-151], t$95$0, N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-151

   1. Initial program 99.3%

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

   if -1.9999999999999999e-151 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.7%

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

      1. flip-- 6.8%

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

      2. add-sqr-sqrt 6.9%

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

      3. div-sub 6.8%

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

      4. pow2 6.8%

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

      5. sub-neg 6.8%

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

      6. add-sqr-sqrt 2.1%

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

      7. hypot-define 2.1%

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

      8. pow2 2.1%

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

      9. sub-neg 2.1%

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

      10. add-sqr-sqrt 2.1%

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

      11. hypot-define 2.1%

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

   4. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} - \frac{{x}^{2} - \varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
   5. Step-by-step derivation

      1. div-sub 2.1%

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

      2. sub-neg 2.1%

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

      3. associate--r+ 45.2%

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

      4. +-inverses 45.2%

         \[\leadsto \frac{\color{blue}{0} - \left(-\varepsilon\right)}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

      5. neg-sub0 45.2%

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

      6. remove-double-neg 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in eps around 0 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

      3. fma-define 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]

      4. associate-*r/ 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \left(\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}}\right)} \]

      5. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon\right)}}{x}\right)} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \varepsilon\right)}{x}\right)} \]

      7. rem-square-sqrt 99.8%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{-1} \cdot \varepsilon\right)}{x}\right)} \]

      8. neg-mul-1 99.8%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left(-\varepsilon\right)}}{x}\right)} \]

      9. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\color{blue}{-0.5 \cdot \varepsilon}}{x}\right)} \]

      10. distribute-frac-neg 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-\frac{0.5 \cdot \varepsilon}{x}}\right)} \]

      11. distribute-frac-neg2 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \varepsilon}{-x}}\right)} \]

      12. neg-mul-1 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \varepsilon}{\color{blue}{-1 \cdot x}}\right)} \]

      13. times-frac 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{-1} \cdot \frac{\varepsilon}{x}}\right)} \]

      14. metadata-eval 99.8%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-0.5} \cdot \frac{\varepsilon}{x}\right)} \]

   9. Simplified 99.8%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, -0.5 \cdot \frac{\varepsilon}{x}\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 99.8%

          \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}} \]

   11. Applied egg-rr 99.8%

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

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.3e-91)
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* -0.5 (/ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.3e-91) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 3.3d-91) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / ((x * 2.0d0) + ((-0.5d0) * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.3e-91) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.3e-91:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.3e-91)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(-0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.3e-91)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.3e-91], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.30000000000000011e-91

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 92.4%

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

      1. neg-mul-1 92.4%

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

   5. Simplified 92.4%

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

   if 3.30000000000000011e-91 < x

   1. Initial program 21.0%

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

      1. flip-- 21.0%

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

      2. add-sqr-sqrt 21.0%

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

      3. div-sub 21.1%

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

      4. pow2 21.1%

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

      5. sub-neg 21.1%

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

      6. add-sqr-sqrt 17.2%

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

      7. hypot-define 17.2%

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

      8. pow2 17.2%

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

      9. sub-neg 17.2%

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

      10. add-sqr-sqrt 17.2%

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

      11. hypot-define 17.2%

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

   4. Applied egg-rr 17.2%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} - \frac{{x}^{2} - \varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
   5. Step-by-step derivation

      1. div-sub 17.2%

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

      2. sub-neg 17.2%

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

      3. associate--r+ 53.5%

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

      4. +-inverses 53.5%

         \[\leadsto \frac{\color{blue}{0} - \left(-\varepsilon\right)}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

      5. neg-sub0 53.5%

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

      6. remove-double-neg 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in eps around 0 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

      3. fma-define 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]

      4. associate-*r/ 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \left(\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}}\right)} \]

      5. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon\right)}}{x}\right)} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \varepsilon\right)}{x}\right)} \]

      7. rem-square-sqrt 86.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{-1} \cdot \varepsilon\right)}{x}\right)} \]

      8. neg-mul-1 86.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left(-\varepsilon\right)}}{x}\right)} \]

      9. distribute-rgt-neg-in 86.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\color{blue}{-0.5 \cdot \varepsilon}}{x}\right)} \]

      10. distribute-frac-neg 86.0%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-\frac{0.5 \cdot \varepsilon}{x}}\right)} \]

      11. distribute-frac-neg2 86.0%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \varepsilon}{-x}}\right)} \]

      12. neg-mul-1 86.0%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \varepsilon}{\color{blue}{-1 \cdot x}}\right)} \]

      13. times-frac 86.0%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{-1} \cdot \frac{\varepsilon}{x}}\right)} \]

      14. metadata-eval 86.0%

          \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-0.5} \cdot \frac{\varepsilon}{x}\right)} \]

   9. Simplified 86.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, -0.5 \cdot \frac{\varepsilon}{x}\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 86.0%

          \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}} \]

   11. Applied egg-rr 86.0%

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

Alternative 4: 44.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

   1. flip-- 65.6%

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

   2. add-sqr-sqrt 65.3%

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

   3. div-sub 65.3%

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

   4. pow2 65.3%

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

   5. sub-neg 65.3%

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

   6. add-sqr-sqrt 63.6%

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

   7. hypot-define 63.6%

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

   8. pow2 63.6%

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

   9. sub-neg 63.6%

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

   10. add-sqr-sqrt 63.6%

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

   11. hypot-define 63.6%

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

4. Applied egg-rr 63.6%

   \[\leadsto \color{blue}{\frac{{x}^{2}}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} - \frac{{x}^{2} - \varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
5. Step-by-step derivation

   1. div-sub 63.6%

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

   2. sub-neg 63.6%

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

   3. associate--r+ 79.6%

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

   4. +-inverses 79.6%

      \[\leadsto \frac{\color{blue}{0} - \left(-\varepsilon\right)}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

   5. neg-sub0 79.6%

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

   6. remove-double-neg 79.6%

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

6. Simplified 79.6%

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

7. Taylor expanded in eps around 0 0.0%

   \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
8. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   2. *-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

   3. fma-define 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]

   4. associate-*r/ 0.0%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \left(\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}}\right)} \]

   5. *-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon\right)}}{x}\right)} \]

   6. unpow2 0.0%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \varepsilon\right)}{x}\right)} \]

   7. rem-square-sqrt 40.7%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{-1} \cdot \varepsilon\right)}{x}\right)} \]

   8. neg-mul-1 40.7%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left(-\varepsilon\right)}}{x}\right)} \]

   9. distribute-rgt-neg-in 40.7%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\color{blue}{-0.5 \cdot \varepsilon}}{x}\right)} \]

   10. distribute-frac-neg 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-\frac{0.5 \cdot \varepsilon}{x}}\right)} \]

   11. distribute-frac-neg2 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \varepsilon}{-x}}\right)} \]

   12. neg-mul-1 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \varepsilon}{\color{blue}{-1 \cdot x}}\right)} \]

   13. times-frac 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{-1} \cdot \frac{\varepsilon}{x}}\right)} \]

   14. metadata-eval 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-0.5} \cdot \frac{\varepsilon}{x}\right)} \]

9. Simplified 40.7%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, -0.5 \cdot \frac{\varepsilon}{x}\right)}} \]
10. Step-by-step derivation

    1. fma-undefine 40.7%

       \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}} \]

11. Applied egg-rr 40.7%

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

Alternative 5: 44.0% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in x around inf 40.2%

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

4. Final simplification 40.2%

   \[\leadsto \frac{\varepsilon}{x} \cdot 0.5 \]
5. Add Preprocessing

Alternative 6: 5.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* x -2.0))

C

double code(double x, double eps) {
	return x * -2.0;
}

Fortran

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

Java

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

Python

def code(x, eps):
	return x * -2.0

Julia

function code(x, eps)
	return Float64(x * -2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = x * -2.0;
end

Wolfram

code[x_, eps_] := N[(x * -2.0), $MachinePrecision]

Derivation

1. Initial program 65.6%

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

   1. flip-- 65.6%

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

   2. add-sqr-sqrt 65.3%

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

   3. div-sub 65.3%

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

   4. pow2 65.3%

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

   5. sub-neg 65.3%

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

   6. add-sqr-sqrt 63.6%

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

   7. hypot-define 63.6%

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

   8. pow2 63.6%

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

   9. sub-neg 63.6%

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

   10. add-sqr-sqrt 63.6%

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

   11. hypot-define 63.6%

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

4. Applied egg-rr 63.6%

   \[\leadsto \color{blue}{\frac{{x}^{2}}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} - \frac{{x}^{2} - \varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
5. Step-by-step derivation

   1. div-sub 63.6%

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

   2. sub-neg 63.6%

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

   3. associate--r+ 79.6%

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

   4. +-inverses 79.6%

      \[\leadsto \frac{\color{blue}{0} - \left(-\varepsilon\right)}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

   5. neg-sub0 79.6%

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

   6. remove-double-neg 79.6%

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

6. Simplified 79.6%

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

7. Taylor expanded in eps around 0 0.0%

   \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
8. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   2. *-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

   3. fma-define 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]

   4. associate-*r/ 0.0%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \left(\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}}\right)} \]

   5. *-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon\right)}}{x}\right)} \]

   6. unpow2 0.0%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \varepsilon\right)}{x}\right)} \]

   7. rem-square-sqrt 40.7%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \left(\color{blue}{-1} \cdot \varepsilon\right)}{x}\right)} \]

   8. neg-mul-1 40.7%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \color{blue}{\left(-\varepsilon\right)}}{x}\right)} \]

   9. distribute-rgt-neg-in 40.7%

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\color{blue}{-0.5 \cdot \varepsilon}}{x}\right)} \]

   10. distribute-frac-neg 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-\frac{0.5 \cdot \varepsilon}{x}}\right)} \]

   11. distribute-frac-neg2 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5 \cdot \varepsilon}{-x}}\right)} \]

   12. neg-mul-1 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{0.5 \cdot \varepsilon}{\color{blue}{-1 \cdot x}}\right)} \]

   13. times-frac 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{-1} \cdot \frac{\varepsilon}{x}}\right)} \]

   14. metadata-eval 40.7%

       \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(x, 2, \color{blue}{-0.5} \cdot \frac{\varepsilon}{x}\right)} \]

9. Simplified 40.7%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x, 2, -0.5 \cdot \frac{\varepsilon}{x}\right)}} \]

10. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

       \[\leadsto \color{blue}{x \cdot -2} \]

12. Simplified 5.3%

    \[\leadsto \color{blue}{x \cdot -2} \]
13. Add Preprocessing

Alternative 7: 3.5% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x

Julia

function code(x, eps)
	return x
end

MATLAB

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

Wolfram

code[x_, eps_] := x

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in x around 0 62.7%

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

   1. neg-mul-1 62.7%

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

5. Simplified 62.7%

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

6. Taylor expanded in x around inf 3.5%

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

Developer target: 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 2024096 
(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
  (/ eps (+ x (sqrt (- (* x x) eps))))

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