ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.7% → 99.5%

Time: 10.0s

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: 62.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x (sqrt (- (pow x 2.0) eps)))))

C

double code(double x, double eps) {
	return eps / (x + sqrt((pow(x, 2.0) - eps)));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return eps / (x + Math.sqrt((Math.pow(x, 2.0) - eps)));
}

Python

def code(x, eps):
	return eps / (x + math.sqrt((math.pow(x, 2.0) - eps)))

Julia

function code(x, eps)
	return Float64(eps / Float64(x + sqrt(Float64((x ^ 2.0) - eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (x + sqrt(((x ^ 2.0) - eps)));
end

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. pow1/2 63.3%

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

4. Applied egg-rr 63.0%

   \[\leadsto x - \color{blue}{{\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{0.5} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.75}\right)}^{0.5}} \]
5. Step-by-step derivation

   1. unpow1/2 63.0%

      \[\leadsto x - \color{blue}{\sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}}} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.75}\right)}^{0.5} \]

   2. unpow1/2 63.0%

      \[\leadsto x - \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \color{blue}{\sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}} \]

6. Simplified 63.0%

   \[\leadsto x - \color{blue}{\sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}} \]
7. Step-by-step derivation

   1. flip-- 62.9%

      \[\leadsto \color{blue}{\frac{x \cdot x - \left(\sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}\right) \cdot \left(\sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}\right)}{x + \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}}} \]

   2. div-inv 62.8%

      \[\leadsto \color{blue}{\left(x \cdot x - \left(\sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}\right) \cdot \left(\sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}\right)\right) \cdot \frac{1}{x + \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.25}} \cdot \sqrt{{\left({x}^{2} - \varepsilon\right)}^{0.75}}}} \]

8. Applied egg-rr 62.8%

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

   1. associate-*r/ 62.8%

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

10. Simplified 99.5%

    \[\leadsto \color{blue}{\frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}}} \]
11. Add Preprocessing

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-151)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (+ x (+ x (/ eps (* x -2.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-151) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-151) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-151:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / (x + (x + (eps / (x * -2.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-151)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(eps / Float64(x * -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -1e-151)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-151], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(eps / N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999994e-152

   1. Initial program 98.5%

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

      1. flip-- 98.3%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 97.7%

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

      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 -9.9999999999999994e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 49.2%

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

      9. hypot-define 49.2%

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

   4. Applied egg-rr 49.2%

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

      1. *-commutative 49.2%

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

      2. +-inverses 49.2%

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

      3. +-lft-identity 49.2%

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

      4. associate-*l/ 49.5%

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

      5. *-lft-identity 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in eps around 0 0.0%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]
   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 98.9%

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

      9. metadata-eval 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-frac 98.9%

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

      12. distribute-rgt-neg-in 98.9%

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

      13. *-commutative 98.9%

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

      14. distribute-lft-neg-in 98.9%

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

      15. fma-define 98.9%

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

      16. distribute-neg-frac 98.9%

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

      17. metadata-eval 98.9%

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

   9. Simplified 98.9%

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

       1. fma-undefine 98.9%

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

       2. clear-num 98.9%

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

       3. associate-*l/ 98.9%

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

       4. *-un-lft-identity 98.9%

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

       5. div-inv 98.9%

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

       6. metadata-eval 98.9%

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

   11. Applied egg-rr 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-151) t_0 (/ eps (+ x (+ x (/ eps (* x -2.0))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-151) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	}
	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 <= (-1d-151)) then
        tmp = t_0
    else
        tmp = eps / (x + (x + (eps / (x * (-2.0d0)))))
    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 <= -1e-151) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-151:
		tmp = t_0
	else:
		tmp = eps / (x + (x + (eps / (x * -2.0))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-151)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(eps / Float64(x * -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-151)
		tmp = t_0;
	else
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	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, -1e-151], t$95$0, N[(eps / N[(x + N[(x + N[(eps / N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999994e-152

   1. Initial program 98.5%

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

   if -9.9999999999999994e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 49.2%

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

      9. hypot-define 49.2%

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

   4. Applied egg-rr 49.2%

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

      1. *-commutative 49.2%

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

      2. +-inverses 49.2%

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

      3. +-lft-identity 49.2%

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

      4. associate-*l/ 49.5%

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

      5. *-lft-identity 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in eps around 0 0.0%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]
   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 98.9%

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

      9. metadata-eval 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-frac 98.9%

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

      12. distribute-rgt-neg-in 98.9%

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

      13. *-commutative 98.9%

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

      14. distribute-lft-neg-in 98.9%

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

      15. fma-define 98.9%

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

      16. distribute-neg-frac 98.9%

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

      17. metadata-eval 98.9%

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

   9. Simplified 98.9%

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

       1. fma-undefine 98.9%

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

       2. clear-num 98.9%

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

       3. associate-*l/ 98.9%

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

       4. *-un-lft-identity 98.9%

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

       5. div-inv 98.9%

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

       6. metadata-eval 98.9%

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

   11. Applied egg-rr 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{-\varepsilon}\\ \mathbf{if}\;x \leq 9 \cdot 10^{-121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- eps)))))
   (if (<= x 9e-121)
     t_0
     (if (<= x 4.8e-89)
       (/ eps (+ x (+ x (/ eps (* x -2.0)))))
       (if (<= x 2.05e-81) t_0 (/ eps (+ (* -0.5 (/ eps x)) (* x 2.0))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(-eps);
	double tmp;
	if (x <= 9e-121) {
		tmp = t_0;
	} else if (x <= 4.8e-89) {
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	} else if (x <= 2.05e-81) {
		tmp = t_0;
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	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 <= 9d-121) then
        tmp = t_0
    else if (x <= 4.8d-89) then
        tmp = eps / (x + (x + (eps / (x * (-2.0d0)))))
    else if (x <= 2.05d-81) then
        tmp = t_0
    else
        tmp = eps / (((-0.5d0) * (eps / x)) + (x * 2.0d0))
    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 <= 9e-121) {
		tmp = t_0;
	} else if (x <= 4.8e-89) {
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	} else if (x <= 2.05e-81) {
		tmp = t_0;
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(-eps)
	tmp = 0
	if x <= 9e-121:
		tmp = t_0
	elif x <= 4.8e-89:
		tmp = eps / (x + (x + (eps / (x * -2.0))))
	elif x <= 2.05e-81:
		tmp = t_0
	else:
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(-eps)))
	tmp = 0.0
	if (x <= 9e-121)
		tmp = t_0;
	elseif (x <= 4.8e-89)
		tmp = Float64(eps / Float64(x + Float64(x + Float64(eps / Float64(x * -2.0)))));
	elseif (x <= 2.05e-81)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(-0.5 * Float64(eps / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(-eps);
	tmp = 0.0;
	if (x <= 9e-121)
		tmp = t_0;
	elseif (x <= 4.8e-89)
		tmp = eps / (x + (x + (eps / (x * -2.0))));
	elseif (x <= 2.05e-81)
		tmp = t_0;
	else
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9e-121], t$95$0, If[LessEqual[x, 4.8e-89], N[(eps / N[(x + N[(x + N[(eps / N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e-81], t$95$0, N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.0000000000000007e-121 or 4.80000000000000032e-89 < x < 2.04999999999999992e-81

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0 96.4%

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

      1. neg-mul-1 96.4%

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

   5. Simplified 96.4%

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

   if 9.0000000000000007e-121 < x < 4.80000000000000032e-89

   1. Initial program 39.3%

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

      1. flip-- 39.3%

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

      2. div-inv 39.3%

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

      3. add-sqr-sqrt 39.2%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 65.1%

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

      9. hypot-define 65.1%

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

   4. Applied egg-rr 65.1%

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

      1. *-commutative 65.1%

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

      2. +-inverses 65.1%

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

      3. +-lft-identity 65.1%

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

      4. associate-*l/ 65.1%

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

      5. *-lft-identity 65.1%

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

   6. Simplified 65.1%

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

   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 70.0%

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

      9. metadata-eval 70.0%

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

      10. metadata-eval 70.0%

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

      11. distribute-neg-frac 70.0%

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

      12. distribute-rgt-neg-in 70.0%

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

      13. *-commutative 70.0%

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

      14. distribute-lft-neg-in 70.0%

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

      15. fma-define 70.0%

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

      16. distribute-neg-frac 70.0%

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

      17. metadata-eval 70.0%

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

   9. Simplified 70.0%

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

       1. fma-undefine 70.0%

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

       2. clear-num 70.0%

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

       3. associate-*l/ 70.0%

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

       4. *-un-lft-identity 70.0%

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

       5. div-inv 70.0%

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

       6. metadata-eval 70.0%

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

   11. Applied egg-rr 70.0%

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

   if 2.04999999999999992e-81 < x

   1. Initial program 19.7%

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

      1. flip-- 19.7%

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

      2. div-inv 19.7%

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

      3. add-sqr-sqrt 19.8%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 59.2%

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

      9. hypot-define 59.2%

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

   4. Applied egg-rr 59.2%

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

      1. *-commutative 59.2%

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

      2. +-inverses 59.2%

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

      3. +-lft-identity 59.2%

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

      4. associate-*l/ 59.4%

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

      5. *-lft-identity 59.4%

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

   6. Simplified 59.4%

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

   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 87.3%

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

      9. metadata-eval 87.3%

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

      10. metadata-eval 87.3%

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

      11. distribute-neg-frac 87.3%

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

      12. distribute-rgt-neg-in 87.3%

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

      13. *-commutative 87.3%

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

      14. distribute-lft-neg-in 87.3%

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

      15. fma-define 87.3%

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

      16. distribute-neg-frac 87.3%

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

      17. metadata-eval 87.3%

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

   9. Simplified 87.3%

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

   10. Taylor expanded in eps around 0 87.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-121}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-81}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. flip-- 63.2%

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

   2. div-inv 63.0%

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

   3. add-sqr-sqrt 62.8%

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

   4. associate--r- 99.3%

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

   5. pow2 99.3%

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

   6. pow2 99.3%

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

   7. sub-neg 99.3%

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

   8. add-sqr-sqrt 79.9%

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

   9. hypot-define 79.9%

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

4. Applied egg-rr 79.9%

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

   1. *-commutative 79.9%

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

   2. +-inverses 79.9%

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

   3. +-lft-identity 79.9%

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

   4. associate-*l/ 80.0%

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

   5. *-lft-identity 80.0%

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

6. Simplified 80.0%

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

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 43.3%

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

   9. metadata-eval 43.3%

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

   10. metadata-eval 43.3%

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

   11. distribute-neg-frac 43.3%

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

   12. distribute-rgt-neg-in 43.3%

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

   13. *-commutative 43.3%

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

   14. distribute-lft-neg-in 43.3%

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

   15. fma-define 43.3%

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

   16. distribute-neg-frac 43.3%

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

   17. metadata-eval 43.3%

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

9. Simplified 43.3%

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

10. Taylor expanded in eps around 0 43.3%

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

11. Final simplification 43.3%

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

Alternative 6: 44.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. flip-- 63.2%

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

   2. div-inv 63.0%

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

   3. add-sqr-sqrt 62.8%

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

   4. associate--r- 99.3%

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

   5. pow2 99.3%

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

   6. pow2 99.3%

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

   7. sub-neg 99.3%

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

   8. add-sqr-sqrt 79.9%

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

   9. hypot-define 79.9%

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

4. Applied egg-rr 79.9%

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

   1. *-commutative 79.9%

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

   2. +-inverses 79.9%

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

   3. +-lft-identity 79.9%

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

   4. associate-*l/ 80.0%

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

   5. *-lft-identity 80.0%

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

6. Simplified 80.0%

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

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 43.3%

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

   9. metadata-eval 43.3%

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

   10. metadata-eval 43.3%

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

   11. distribute-neg-frac 43.3%

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

   12. distribute-rgt-neg-in 43.3%

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

   13. *-commutative 43.3%

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

   14. distribute-lft-neg-in 43.3%

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

   15. fma-define 43.3%

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

   16. distribute-neg-frac 43.3%

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

   17. metadata-eval 43.3%

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

9. Simplified 43.3%

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

    1. fma-undefine 43.3%

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

    2. clear-num 43.3%

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

    3. associate-*l/ 43.3%

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

    4. *-un-lft-identity 43.3%

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

    5. div-inv 43.3%

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

    6. metadata-eval 43.3%

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

11. Applied egg-rr 43.3%

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

12. Final simplification 43.3%

    \[\leadsto \frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x \cdot -2}\right)} \]
13. Add Preprocessing

Alternative 7: 43.7% 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 63.3%

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

3. Taylor expanded in x around inf 42.8%

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

4. Final simplification 42.8%

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

Alternative 8: 5.4% 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 63.3%

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

   1. flip-- 63.2%

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

   2. div-inv 63.0%

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

   3. add-sqr-sqrt 62.8%

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

   4. associate--r- 99.3%

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

   5. pow2 99.3%

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

   6. pow2 99.3%

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

   7. sub-neg 99.3%

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

   8. add-sqr-sqrt 79.9%

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

   9. hypot-define 79.9%

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

4. Applied egg-rr 79.9%

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

   1. *-commutative 79.9%

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

   2. +-inverses 79.9%

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

   3. +-lft-identity 79.9%

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

   4. associate-*l/ 80.0%

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

   5. *-lft-identity 80.0%

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

6. Simplified 80.0%

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

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 43.3%

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

   9. metadata-eval 43.3%

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

   10. metadata-eval 43.3%

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

   11. distribute-neg-frac 43.3%

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

   12. distribute-rgt-neg-in 43.3%

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

   13. *-commutative 43.3%

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

   14. distribute-lft-neg-in 43.3%

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

   15. fma-define 43.3%

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

   16. distribute-neg-frac 43.3%

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

   17. metadata-eval 43.3%

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

9. Simplified 43.3%

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

10. Taylor expanded in eps around inf 5.4%

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

    1. *-commutative 5.4%

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

12. Simplified 5.4%

    \[\leadsto \color{blue}{x \cdot -2} \]
13. 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 2024103 
(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))))