ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.2% → 98.9%

Time: 9.4s

Alternatives: 8

Speedup: 0.9×

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: 61.2% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-153)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (fma x 2.0 (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / fma(x, 2.0, (eps * (-0.5 / x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -4e-153)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / fma(x, 2.0, Float64(eps * Float64(-0.5 / x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -4e-153], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x * 2.0 + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 99.0%

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

      1. flip-- 98.9%

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

      2. div-inv 98.6%

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

      3. add-sqr-sqrt 98.4%

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

      4. associate--r- 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. sub-neg 99.2%

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

      8. add-sqr-sqrt 99.2%

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

      9. hypot-define 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. +-inverses 99.2%

         \[\leadsto \frac{1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \cdot \left(\color{blue}{0} + \varepsilon\right) \]

      3. +-lft-identity 99.2%

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

      4. associate-*l/ 99.2%

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

      5. *-lft-identity 99.2%

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

   6. Simplified 99.2%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 11.1%

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

      1. flip-- 11.0%

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

      2. div-inv 11.1%

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

      3. add-sqr-sqrt 11.3%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 46.7%

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

      9. hypot-define 46.7%

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

   4. Applied egg-rr 46.7%

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

      1. *-commutative 46.7%

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

      2. +-inverses 46.7%

         \[\leadsto \frac{1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \cdot \left(\color{blue}{0} + \varepsilon\right) \]

      3. +-lft-identity 46.7%

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

      4. associate-*l/ 46.9%

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

      5. *-lft-identity 46.9%

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

   6. Simplified 46.9%

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

   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-/l* 0.0%

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

      5. associate-*r* 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-*r* 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 97.5%

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

      11. metadata-eval 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-153)
   (- x (hypot (sqrt (- eps)) x))
   (/ eps (fma x 2.0 (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / fma(x, 2.0, (eps * (-0.5 / x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -4e-153)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / fma(x, 2.0, Float64(eps * Float64(-0.5 / x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -4e-153], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x * 2.0 + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 99.0%

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

      1. sub-neg 99.0%

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

      2. +-commutative 99.0%

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

      3. add-sqr-sqrt 99.0%

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

      4. hypot-define 99.0%

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

   4. Applied egg-rr 99.0%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 11.1%

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

      1. flip-- 11.0%

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

      2. div-inv 11.1%

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

      3. add-sqr-sqrt 11.3%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 46.7%

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

      9. hypot-define 46.7%

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

   4. Applied egg-rr 46.7%

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

      1. *-commutative 46.7%

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

      2. +-inverses 46.7%

         \[\leadsto \frac{1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \cdot \left(\color{blue}{0} + \varepsilon\right) \]

      3. +-lft-identity 46.7%

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

      4. associate-*l/ 46.9%

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

      5. *-lft-identity 46.9%

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

   6. Simplified 46.9%

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

   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-/l* 0.0%

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

      5. associate-*r* 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-*r* 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 97.5%

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

      11. metadata-eval 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{fma}\left(\frac{-0.5}{x}, \varepsilon, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-153) t_0 (/ eps (+ x (fma (/ -0.5 x) eps x))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x + fma((-0.5 / x), 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 <= -4e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + fma(Float64(-0.5 / x), eps, x)));
	end
	return 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-153], t$95$0, N[(eps / N[(x + N[(N[(-0.5 / x), $MachinePrecision] * eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 99.0%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 11.1%

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

      1. flip-- 11.0%

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

      2. div-inv 11.1%

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

      3. add-sqr-sqrt 11.3%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 46.7%

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

      9. hypot-define 46.7%

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

   4. Applied egg-rr 46.7%

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

      1. *-commutative 46.7%

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

      2. +-inverses 46.7%

         \[\leadsto \frac{1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \cdot \left(\color{blue}{0} + \varepsilon\right) \]

      3. +-lft-identity 46.7%

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

      4. associate-*l/ 46.9%

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

      5. *-lft-identity 46.9%

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

   6. Simplified 46.9%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/l* 0.0%

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

      4. associate-*r* 0.0%

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

      5. *-commutative 0.0%

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

      6. associate-*r/ 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 97.5%

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

      9. metadata-eval 97.5%

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

      10. metadata-eval 97.5%

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

      11. distribute-neg-frac 97.5%

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

      12. distribute-rgt-neg-in 97.5%

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

      13. *-commutative 97.5%

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

      14. distribute-lft-neg-in 97.5%

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

      15. fma-define 97.5%

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

      16. distribute-neg-frac 97.5%

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

      17. metadata-eval 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{fma}\left(\frac{-0.5}{x}, \varepsilon, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-153) t_0 (/ eps (fma x 2.0 (* eps (/ -0.5 x)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-153) {
		tmp = t_0;
	} else {
		tmp = eps / fma(x, 2.0, (eps * (-0.5 / x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -4e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / fma(x, 2.0, Float64(eps * Float64(-0.5 / x))));
	end
	return 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-153], t$95$0, N[(eps / N[(x * 2.0 + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 99.0%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 11.1%

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

      1. flip-- 11.0%

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

      2. div-inv 11.1%

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

      3. add-sqr-sqrt 11.3%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 46.7%

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

      9. hypot-define 46.7%

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

   4. Applied egg-rr 46.7%

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

      1. *-commutative 46.7%

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

      2. +-inverses 46.7%

         \[\leadsto \frac{1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \cdot \left(\color{blue}{0} + \varepsilon\right) \]

      3. +-lft-identity 46.7%

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

      4. associate-*l/ 46.9%

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

      5. *-lft-identity 46.9%

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

   6. Simplified 46.9%

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

   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-/l* 0.0%

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

      5. associate-*r* 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-*r* 0.0%

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

      8. associate-*r/ 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 97.5%

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

      11. metadata-eval 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{0.5 + \frac{\frac{\varepsilon \cdot 0.125}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-153) t_0 (* eps (/ (+ 0.5 (/ (/ (* eps 0.125) x) x)) x)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-153) {
		tmp = t_0;
	} else {
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / 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 <= (-4d-153)) then
        tmp = t_0
    else
        tmp = eps * ((0.5d0 + (((eps * 0.125d0) / x) / x)) / 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 <= -4e-153) {
		tmp = t_0;
	} else {
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -4e-153:
		tmp = t_0
	else:
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -4e-153)
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(Float64(0.5 + Float64(Float64(Float64(eps * 0.125) / x) / x)) / 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 <= -4e-153)
		tmp = t_0;
	else
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / 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, -4e-153], t$95$0, N[(eps * N[(N[(0.5 + N[(N[(N[(eps * 0.125), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 99.0%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 11.1%

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

   3. Taylor expanded in eps around 0 88.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(0.125 \cdot \frac{\varepsilon}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)} \]

   4. Taylor expanded in x around inf 97.1%

      \[\leadsto \varepsilon \cdot \color{blue}{\frac{0.5 + 0.125 \cdot \frac{\varepsilon}{{x}^{2}}}{x}} \]
   5. Step-by-step derivation

      1. associate-*r/ 97.1%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\frac{0.125 \cdot \varepsilon}{{x}^{2}}}}{x} \]

      2. *-commutative 97.1%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \frac{\color{blue}{\varepsilon \cdot 0.125}}{{x}^{2}}}{x} \]

      3. associate-/l* 97.1%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\varepsilon \cdot \frac{0.125}{{x}^{2}}}}{x} \]

   6. Simplified 97.1%

      \[\leadsto \varepsilon \cdot \color{blue}{\frac{0.5 + \varepsilon \cdot \frac{0.125}{{x}^{2}}}{x}} \]
   7. Step-by-step derivation

      1. associate-*r/ 97.1%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\frac{\varepsilon \cdot 0.125}{{x}^{2}}}}{x} \]

      2. unpow2 97.1%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \frac{\varepsilon \cdot 0.125}{\color{blue}{x \cdot x}}}{x} \]

      3. associate-/r* 97.1%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\frac{\frac{\varepsilon \cdot 0.125}{x}}{x}}}{x} \]

   8. Applied egg-rr 97.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{0.5 + \frac{\frac{\varepsilon \cdot 0.125}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{-\varepsilon}\\ \mathbf{if}\;x \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;\varepsilon \cdot \frac{0.5 + \frac{\frac{\varepsilon \cdot 0.125}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- eps)))))
   (if (<= x 2.15e-94)
     t_0
     (if (<= x 1.45e-68)
       (* eps (/ (+ 0.5 (/ (/ (* eps 0.125) x) x)) x))
       (if (<= x 3.3e-66) t_0 (* 0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(-eps);
	double tmp;
	if (x <= 2.15e-94) {
		tmp = t_0;
	} else if (x <= 1.45e-68) {
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / x);
	} else if (x <= 3.3e-66) {
		tmp = t_0;
	} else {
		tmp = 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(-eps)
    if (x <= 2.15d-94) then
        tmp = t_0
    else if (x <= 1.45d-68) then
        tmp = eps * ((0.5d0 + (((eps * 0.125d0) / x) / x)) / x)
    else if (x <= 3.3d-66) then
        tmp = t_0
    else
        tmp = 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(-eps);
	double tmp;
	if (x <= 2.15e-94) {
		tmp = t_0;
	} else if (x <= 1.45e-68) {
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / x);
	} else if (x <= 3.3e-66) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(-eps)
	tmp = 0
	if x <= 2.15e-94:
		tmp = t_0
	elif x <= 1.45e-68:
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / x)
	elif x <= 3.3e-66:
		tmp = t_0
	else:
		tmp = 0.5 * (eps / x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(-eps)))
	tmp = 0.0
	if (x <= 2.15e-94)
		tmp = t_0;
	elseif (x <= 1.45e-68)
		tmp = Float64(eps * Float64(Float64(0.5 + Float64(Float64(Float64(eps * 0.125) / x) / x)) / x));
	elseif (x <= 3.3e-66)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(eps / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(-eps);
	tmp = 0.0;
	if (x <= 2.15e-94)
		tmp = t_0;
	elseif (x <= 1.45e-68)
		tmp = eps * ((0.5 + (((eps * 0.125) / x) / x)) / x);
	elseif (x <= 3.3e-66)
		tmp = t_0;
	else
		tmp = 0.5 * (eps / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.15e-94], t$95$0, If[LessEqual[x, 1.45e-68], N[(eps * N[(N[(0.5 + N[(N[(N[(eps * 0.125), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-66], t$95$0, N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.1499999999999999e-94 or 1.45e-68 < x < 3.2999999999999999e-66

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0 91.6%

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

      1. neg-mul-1 91.6%

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

   5. Simplified 91.6%

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

   if 2.1499999999999999e-94 < x < 1.45e-68

   1. Initial program 43.1%

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

   3. Taylor expanded in eps around 0 69.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(0.125 \cdot \frac{\varepsilon}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)} \]

   4. Taylor expanded in x around inf 69.6%

      \[\leadsto \varepsilon \cdot \color{blue}{\frac{0.5 + 0.125 \cdot \frac{\varepsilon}{{x}^{2}}}{x}} \]
   5. Step-by-step derivation

      1. associate-*r/ 69.6%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\frac{0.125 \cdot \varepsilon}{{x}^{2}}}}{x} \]

      2. *-commutative 69.6%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \frac{\color{blue}{\varepsilon \cdot 0.125}}{{x}^{2}}}{x} \]

      3. associate-/l* 69.6%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\varepsilon \cdot \frac{0.125}{{x}^{2}}}}{x} \]

   6. Simplified 69.6%

      \[\leadsto \varepsilon \cdot \color{blue}{\frac{0.5 + \varepsilon \cdot \frac{0.125}{{x}^{2}}}{x}} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.6%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\frac{\varepsilon \cdot 0.125}{{x}^{2}}}}{x} \]

      2. unpow2 69.6%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \frac{\varepsilon \cdot 0.125}{\color{blue}{x \cdot x}}}{x} \]

      3. associate-/r* 69.6%

         \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\frac{\frac{\varepsilon \cdot 0.125}{x}}{x}}}{x} \]

   8. Applied egg-rr 69.6%

      \[\leadsto \varepsilon \cdot \frac{0.5 + \color{blue}{\frac{\frac{\varepsilon \cdot 0.125}{x}}{x}}}{x} \]

   if 3.2999999999999999e-66 < x

   1. Initial program 22.1%

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

   3. Taylor expanded in x around inf 84.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;\varepsilon \cdot \frac{0.5 + \frac{\frac{\varepsilon \cdot 0.125}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-66}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.1% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.4%

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

3. Taylor expanded in x around inf 41.3%

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

4. Final simplification 41.3%

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

Alternative 8: 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 65.4%

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

   1. sub-neg 65.4%

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

   2. +-commutative 65.4%

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

   3. add-sqr-sqrt 64.9%

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

   4. distribute-rgt-neg-in 64.9%

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

   5. fma-define 64.9%

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

   6. pow1/2 64.9%

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

   7. sqrt-pow1 64.9%

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

   8. pow2 64.9%

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

   9. metadata-eval 64.9%

      \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.25}}, -\sqrt{\sqrt{x \cdot x - \varepsilon}}, x\right) \]

   10. pow1/2 64.9%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -\sqrt{\color{blue}{{\left(x \cdot x - \varepsilon\right)}^{0.5}}}, x\right) \]

   11. sqrt-pow1 64.8%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -\color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(\frac{0.5}{2}\right)}}, x\right) \]

   12. pow2 64.8%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -{\left(\color{blue}{{x}^{2}} - \varepsilon\right)}^{\left(\frac{0.5}{2}\right)}, x\right) \]

   13. metadata-eval 64.8%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -{\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.25}}, x\right) \]

4. Applied egg-rr 64.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -{\left({x}^{2} - \varepsilon\right)}^{0.25}, x\right)} \]

5. Taylor expanded in eps around 0 4.1%

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

   1. distribute-rgt1-in 4.1%

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

   2. metadata-eval 4.1%

      \[\leadsto \color{blue}{0} \cdot x \]

   3. mul0-lft 4.1%

      \[\leadsto \color{blue}{0} \]

7. Simplified 4.1%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 4.1%

   \[\leadsto 0 \]
9. 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 2024059 
(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))))