ENA, Section 1.4, Exercise 4d

Percentage Accurate: 79.4% → 86.9%

Time: 11.9s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-195} \lor \neg \left(\varepsilon \leq 1.36 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8e-195) (not (<= eps 1.36e-163)))
   (/ eps (+ x (sqrt (- (pow x 2.0) eps))))
   (- x (sqrt (- (* x x) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8e-195) || !(eps <= 1.36e-163)) {
		tmp = eps / (x + sqrt((pow(x, 2.0) - eps)));
	} else {
		tmp = x - sqrt(((x * x) - eps));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-8d-195)) .or. (.not. (eps <= 1.36d-163))) then
        tmp = eps / (x + sqrt(((x ** 2.0d0) - eps)))
    else
        tmp = x - sqrt(((x * x) - eps))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -8e-195) || !(eps <= 1.36e-163)) {
		tmp = eps / (x + Math.sqrt((Math.pow(x, 2.0) - eps)));
	} else {
		tmp = x - Math.sqrt(((x * x) - eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -8e-195) or not (eps <= 1.36e-163):
		tmp = eps / (x + math.sqrt((math.pow(x, 2.0) - eps)))
	else:
		tmp = x - math.sqrt(((x * x) - eps))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8e-195) || !(eps <= 1.36e-163))
		tmp = Float64(eps / Float64(x + sqrt(Float64((x ^ 2.0) - eps))));
	else
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -8e-195) || ~((eps <= 1.36e-163)))
		tmp = eps / (x + sqrt(((x ^ 2.0) - eps)));
	else
		tmp = x - sqrt(((x * x) - eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8e-195], N[Not[LessEqual[eps, 1.36e-163]], $MachinePrecision]], N[(eps / N[(x + N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.0000000000000007e-195 or 1.36e-163 < eps

   1. Initial program 76.6%

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

      1. flip-- 76.6%

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

      2. div-inv 76.4%

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

      3. add-sqr-sqrt 76.4%

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

      4. associate--r- 93.5%

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

      5. pow2 93.5%

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

      6. pow2 93.5%

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

      7. sub-neg 93.5%

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

      8. add-sqr-sqrt 81.1%

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

      9. hypot-def 81.1%

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

   4. Applied egg-rr 81.1%

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

      1. +-inverses 81.1%

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

      2. +-lft-identity 81.1%

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

      3. associate-*r/ 81.1%

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

      4. associate-/l* 81.1%

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

      5. /-rgt-identity 81.1%

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

   6. Simplified 81.1%

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

      1. hypot-udef 81.1%

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

      2. unpow2 81.1%

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

      3. add-sqr-sqrt 93.6%

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

      4. sub-neg 93.6%

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

      5. pow1/2 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. unpow1/2 93.6%

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

   10. Simplified 93.6%

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

   if -8.0000000000000007e-195 < eps < 1.36e-163

   1. Initial program 81.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-195} \lor \neg \left(\varepsilon \leq 1.36 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-163}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.8e-195)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (if (<= eps 6.2e-163)
     (- x (sqrt (- (* x x) eps)))
     (/ eps (+ (/ eps (/ x -0.5)) (* x 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.8e-195) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else if (eps <= 6.2e-163) {
		tmp = x - sqrt(((x * x) - eps));
	} else {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -4.8e-195) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else if (eps <= 6.2e-163) {
		tmp = x - Math.sqrt(((x * x) - eps));
	} else {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -4.8e-195:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	elif eps <= 6.2e-163:
		tmp = x - math.sqrt(((x * x) - eps))
	else:
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4.8e-195)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	elseif (eps <= 6.2e-163)
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	else
		tmp = Float64(eps / Float64(Float64(eps / Float64(x / -0.5)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -4.8e-195)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	elseif (eps <= 6.2e-163)
		tmp = x - sqrt(((x * x) - eps));
	else
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4.8e-195], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6.2e-163], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(eps / N[(x / -0.5), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.8e-195

   1. Initial program 84.7%

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

      1. flip-- 84.6%

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

      2. div-inv 84.4%

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

      3. add-sqr-sqrt 84.2%

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

      4. associate--r- 96.2%

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

      5. pow2 96.2%

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

      6. pow2 96.2%

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

      7. sub-neg 96.2%

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

      8. add-sqr-sqrt 96.2%

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

      9. hypot-def 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. +-inverses 96.2%

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

      2. +-lft-identity 96.2%

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

      3. associate-*r/ 96.3%

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

      4. associate-/l* 96.3%

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

      5. /-rgt-identity 96.3%

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

   6. Simplified 96.3%

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

   if -4.8e-195 < eps < 6.19999999999999949e-163

   1. Initial program 81.8%

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

   if 6.19999999999999949e-163 < eps

   1. Initial program 33.5%

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

      1. flip-- 33.6%

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

      2. div-inv 33.6%

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

      3. add-sqr-sqrt 34.8%

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

      4. associate--r- 78.8%

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

      5. pow2 78.8%

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

      6. pow2 78.8%

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

      7. sub-neg 78.8%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. +-lft-identity 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. /-rgt-identity 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 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-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 77.6%

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

      8. associate-*r* 77.6%

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

      9. metadata-eval 77.6%

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

      10. associate-*r/ 77.6%

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

      11. *-commutative 77.6%

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

   9. Simplified 77.6%

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

       1. fma-udef 77.6%

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

       2. +-commutative 77.6%

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

       3. associate-*l/ 77.6%

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

       4. associate-/l* 77.6%

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

   11. Applied egg-rr 77.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-163}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{-\varepsilon}\\ t_1 := \frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 30:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \mathbf{elif}\;x \leq 54000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- eps))))
        (t_1 (/ eps (+ (/ eps (/ x -0.5)) (* x 2.0)))))
   (if (<= x 1.9e-114)
     t_0
     (if (<= x 3.2e-91)
       t_1
       (if (<= x 1.65e-65)
         t_0
         (if (<= x 30.0)
           (- x (+ x (* -0.5 (/ eps x))))
           (if (<= x 54000000.0) t_1 (- x x))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(-eps);
	double t_1 = eps / ((eps / (x / -0.5)) + (x * 2.0));
	double tmp;
	if (x <= 1.9e-114) {
		tmp = t_0;
	} else if (x <= 3.2e-91) {
		tmp = t_1;
	} else if (x <= 1.65e-65) {
		tmp = t_0;
	} else if (x <= 30.0) {
		tmp = x - (x + (-0.5 * (eps / x)));
	} else if (x <= 54000000.0) {
		tmp = t_1;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = x - sqrt(-eps)
    t_1 = eps / ((eps / (x / (-0.5d0))) + (x * 2.0d0))
    if (x <= 1.9d-114) then
        tmp = t_0
    else if (x <= 3.2d-91) then
        tmp = t_1
    else if (x <= 1.65d-65) then
        tmp = t_0
    else if (x <= 30.0d0) then
        tmp = x - (x + ((-0.5d0) * (eps / x)))
    else if (x <= 54000000.0d0) then
        tmp = t_1
    else
        tmp = x - x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(-eps);
	double t_1 = eps / ((eps / (x / -0.5)) + (x * 2.0));
	double tmp;
	if (x <= 1.9e-114) {
		tmp = t_0;
	} else if (x <= 3.2e-91) {
		tmp = t_1;
	} else if (x <= 1.65e-65) {
		tmp = t_0;
	} else if (x <= 30.0) {
		tmp = x - (x + (-0.5 * (eps / x)));
	} else if (x <= 54000000.0) {
		tmp = t_1;
	} else {
		tmp = x - x;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(-eps)
	t_1 = eps / ((eps / (x / -0.5)) + (x * 2.0))
	tmp = 0
	if x <= 1.9e-114:
		tmp = t_0
	elif x <= 3.2e-91:
		tmp = t_1
	elif x <= 1.65e-65:
		tmp = t_0
	elif x <= 30.0:
		tmp = x - (x + (-0.5 * (eps / x)))
	elif x <= 54000000.0:
		tmp = t_1
	else:
		tmp = x - x
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(-eps)))
	t_1 = Float64(eps / Float64(Float64(eps / Float64(x / -0.5)) + Float64(x * 2.0)))
	tmp = 0.0
	if (x <= 1.9e-114)
		tmp = t_0;
	elseif (x <= 3.2e-91)
		tmp = t_1;
	elseif (x <= 1.65e-65)
		tmp = t_0;
	elseif (x <= 30.0)
		tmp = Float64(x - Float64(x + Float64(-0.5 * Float64(eps / x))));
	elseif (x <= 54000000.0)
		tmp = t_1;
	else
		tmp = Float64(x - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(-eps);
	t_1 = eps / ((eps / (x / -0.5)) + (x * 2.0));
	tmp = 0.0;
	if (x <= 1.9e-114)
		tmp = t_0;
	elseif (x <= 3.2e-91)
		tmp = t_1;
	elseif (x <= 1.65e-65)
		tmp = t_0;
	elseif (x <= 30.0)
		tmp = x - (x + (-0.5 * (eps / x)));
	elseif (x <= 54000000.0)
		tmp = t_1;
	else
		tmp = x - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps / N[(N[(eps / N[(x / -0.5), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9e-114], t$95$0, If[LessEqual[x, 3.2e-91], t$95$1, If[LessEqual[x, 1.65e-65], t$95$0, If[LessEqual[x, 30.0], N[(x - N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 54000000.0], t$95$1, N[(x - x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.8999999999999999e-114 or 3.19999999999999996e-91 < x < 1.6500000000000001e-65

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0 86.8%

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

      1. neg-mul-1 86.8%

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

   5. Simplified 86.8%

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

   if 1.8999999999999999e-114 < x < 3.19999999999999996e-91 or 30 < x < 5.4e7

   1. Initial program 38.1%

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

      1. flip-- 38.1%

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

      2. div-inv 38.0%

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

      3. add-sqr-sqrt 38.1%

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

      4. associate--r- 88.0%

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

      5. pow2 88.0%

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

      6. pow2 88.0%

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

      7. sub-neg 88.0%

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

      8. add-sqr-sqrt 44.5%

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

      9. hypot-def 44.5%

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

   4. Applied egg-rr 44.5%

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

      1. +-inverses 44.5%

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

      2. +-lft-identity 44.5%

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

      3. associate-*r/ 44.6%

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

      4. associate-/l* 44.6%

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

      5. /-rgt-identity 44.6%

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

   6. Simplified 44.6%

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

   7. Taylor expanded in x around inf 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-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 70.3%

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

      8. associate-*r* 70.3%

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

      9. metadata-eval 70.3%

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

      10. associate-*r/ 70.3%

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

      11. *-commutative 70.3%

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

   9. Simplified 70.3%

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

       1. fma-udef 70.3%

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

       2. +-commutative 70.3%

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

       3. associate-*l/ 70.3%

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

       4. associate-/l* 70.3%

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

   11. Applied egg-rr 70.3%

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

   if 1.6500000000000001e-65 < x < 30

   1. Initial program 69.0%

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

   3. Taylor expanded in x around inf 57.7%

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

   if 5.4e7 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto x - \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-65}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{elif}\;x \leq 30:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \mathbf{elif}\;x \leq 54000000:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.5 \cdot 10^{-164}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2.5e-164)
   (- x (sqrt (- (* x x) eps)))
   (/ eps (+ (/ eps (/ x -0.5)) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2.5e-164) {
		tmp = x - sqrt(((x * x) - eps));
	} else {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 2.5d-164) then
        tmp = x - sqrt(((x * x) - eps))
    else
        tmp = eps / ((eps / (x / (-0.5d0))) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2.5e-164) {
		tmp = x - Math.sqrt(((x * x) - eps));
	} else {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2.5e-164:
		tmp = x - math.sqrt(((x * x) - eps))
	else:
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2.5e-164)
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	else
		tmp = Float64(eps / Float64(Float64(eps / Float64(x / -0.5)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2.5e-164)
		tmp = x - sqrt(((x * x) - eps));
	else
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2.5e-164], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(eps / N[(x / -0.5), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.49999999999999981e-164

   1. Initial program 83.3%

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

   if 2.49999999999999981e-164 < eps

   1. Initial program 33.5%

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

      1. flip-- 33.6%

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

      2. div-inv 33.6%

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

      3. add-sqr-sqrt 34.8%

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

      4. associate--r- 78.8%

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

      5. pow2 78.8%

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

      6. pow2 78.8%

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

      7. sub-neg 78.8%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. +-lft-identity 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. /-rgt-identity 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 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-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 77.6%

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

      8. associate-*r* 77.6%

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

      9. metadata-eval 77.6%

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

      10. associate-*r/ 77.6%

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

      11. *-commutative 77.6%

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

   9. Simplified 77.6%

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

       1. fma-udef 77.6%

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

       2. +-commutative 77.6%

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

       3. associate-*l/ 77.6%

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

       4. associate-/l* 77.6%

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

   11. Applied egg-rr 77.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.5 \cdot 10^{-164}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-297}:\\ \;\;\;\;\frac{\varepsilon}{x + \sqrt{-\varepsilon}}\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-166}:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.75e-297)
   (/ eps (+ x (sqrt (- eps))))
   (if (<= eps 9.5e-166)
     (- x (+ x (* -0.5 (/ eps x))))
     (/ eps (+ (/ eps (/ x -0.5)) (* x 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.75e-297) {
		tmp = eps / (x + sqrt(-eps));
	} else if (eps <= 9.5e-166) {
		tmp = x - (x + (-0.5 * (eps / x)));
	} else {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.75d-297)) then
        tmp = eps / (x + sqrt(-eps))
    else if (eps <= 9.5d-166) then
        tmp = x - (x + ((-0.5d0) * (eps / x)))
    else
        tmp = eps / ((eps / (x / (-0.5d0))) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.75e-297) {
		tmp = eps / (x + Math.sqrt(-eps));
	} else if (eps <= 9.5e-166) {
		tmp = x - (x + (-0.5 * (eps / x)));
	} else {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.75e-297:
		tmp = eps / (x + math.sqrt(-eps))
	elif eps <= 9.5e-166:
		tmp = x - (x + (-0.5 * (eps / x)))
	else:
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.75e-297)
		tmp = Float64(eps / Float64(x + sqrt(Float64(-eps))));
	elseif (eps <= 9.5e-166)
		tmp = Float64(x - Float64(x + Float64(-0.5 * Float64(eps / x))));
	else
		tmp = Float64(eps / Float64(Float64(eps / Float64(x / -0.5)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.75e-297)
		tmp = eps / (x + sqrt(-eps));
	elseif (eps <= 9.5e-166)
		tmp = x - (x + (-0.5 * (eps / x)));
	else
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.75e-297], N[(eps / N[(x + N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 9.5e-166], N[(x - N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(eps / N[(x / -0.5), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.7499999999999999e-297

   1. Initial program 85.9%

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

      1. flip-- 85.8%

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

      2. div-inv 85.6%

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

      3. add-sqr-sqrt 85.4%

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

      4. associate--r- 89.8%

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

      5. pow2 89.8%

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

      6. pow2 89.8%

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

      7. sub-neg 89.8%

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

      8. add-sqr-sqrt 89.8%

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

      9. hypot-def 89.8%

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

   4. Applied egg-rr 89.8%

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

      1. +-inverses 89.8%

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

      2. +-lft-identity 89.8%

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

      3. associate-*r/ 89.9%

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

      4. associate-/l* 89.9%

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

      5. /-rgt-identity 89.9%

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

   6. Simplified 89.9%

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

      1. hypot-udef 89.9%

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

      2. unpow2 89.9%

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

      3. add-sqr-sqrt 89.9%

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

      4. sub-neg 89.9%

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

      5. pow1/2 89.9%

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

   8. Applied egg-rr 89.9%

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

      1. unpow1/2 89.9%

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

   10. Simplified 89.9%

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

   11. Taylor expanded in x around 0 73.9%

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

       1. neg-mul-1 73.9%

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

   13. Simplified 73.9%

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

   if -1.7499999999999999e-297 < eps < 9.50000000000000046e-166

   1. Initial program 72.7%

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

   3. Taylor expanded in x around inf 68.4%

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

   if 9.50000000000000046e-166 < eps

   1. Initial program 33.5%

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

      1. flip-- 33.6%

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

      2. div-inv 33.6%

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

      3. add-sqr-sqrt 34.8%

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

      4. associate--r- 78.8%

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

      5. pow2 78.8%

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

      6. pow2 78.8%

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

      7. sub-neg 78.8%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. +-lft-identity 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. /-rgt-identity 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 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-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 77.6%

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

      8. associate-*r* 77.6%

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

      9. metadata-eval 77.6%

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

      10. associate-*r/ 77.6%

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

      11. *-commutative 77.6%

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

   9. Simplified 77.6%

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

       1. fma-udef 77.6%

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

       2. +-commutative 77.6%

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

       3. associate-*l/ 77.6%

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

       4. associate-/l* 77.6%

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

   11. Applied egg-rr 77.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-297}:\\ \;\;\;\;\frac{\varepsilon}{x + \sqrt{-\varepsilon}}\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-166}:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1 \cdot 10^{-203} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1e-203) (not (<= eps 1.15e-163)))
   (/ eps (+ (/ eps (/ x -0.5)) (* x 2.0)))
   (- x (+ x (* -0.5 (/ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1e-203) || !(eps <= 1.15e-163)) {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	} else {
		tmp = x - (x + (-0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1d-203)) .or. (.not. (eps <= 1.15d-163))) then
        tmp = eps / ((eps / (x / (-0.5d0))) + (x * 2.0d0))
    else
        tmp = x - (x + ((-0.5d0) * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1e-203) || !(eps <= 1.15e-163)) {
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	} else {
		tmp = x - (x + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1e-203) or not (eps <= 1.15e-163):
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0))
	else:
		tmp = x - (x + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1e-203) || !(eps <= 1.15e-163))
		tmp = Float64(eps / Float64(Float64(eps / Float64(x / -0.5)) + Float64(x * 2.0)));
	else
		tmp = Float64(x - Float64(x + Float64(-0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1e-203) || ~((eps <= 1.15e-163)))
		tmp = eps / ((eps / (x / -0.5)) + (x * 2.0));
	else
		tmp = x - (x + (-0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1e-203], N[Not[LessEqual[eps, 1.15e-163]], $MachinePrecision]], N[(eps / N[(N[(eps / N[(x / -0.5), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1e-203 or 1.15e-163 < eps

   1. Initial program 77.4%

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

      1. flip-- 77.3%

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

      2. div-inv 77.1%

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

      3. add-sqr-sqrt 77.2%

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

      4. associate--r- 93.7%

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

      5. pow2 93.7%

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

      6. pow2 93.7%

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

      7. sub-neg 93.7%

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

      8. add-sqr-sqrt 81.7%

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

      9. hypot-def 81.7%

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

   4. Applied egg-rr 81.7%

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

      1. +-inverses 81.7%

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

      2. +-lft-identity 81.7%

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

      3. associate-*r/ 81.7%

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

      4. associate-/l* 81.7%

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

      5. /-rgt-identity 81.7%

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

   6. Simplified 81.7%

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

   7. Taylor expanded in x around inf 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-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 29.9%

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

      8. associate-*r* 29.9%

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

      9. metadata-eval 29.9%

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

      10. associate-*r/ 29.9%

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

      11. *-commutative 29.9%

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

   9. Simplified 29.9%

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

       1. fma-udef 29.9%

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

       2. +-commutative 29.9%

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

       3. associate-*l/ 29.9%

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

       4. associate-/l* 29.9%

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

   11. Applied egg-rr 29.9%

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

   if -1e-203 < eps < 1.15e-163

   1. Initial program 81.0%

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

   3. Taylor expanded in x around inf 46.9%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1 \cdot 10^{-203} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{\frac{x}{-0.5}} + x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-195} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -7.5e-195) (not (<= eps 1.15e-164)))
   (* (/ eps x) 0.5)
   (- x (+ x (* -0.5 (/ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.5e-195) || !(eps <= 1.15e-164)) {
		tmp = (eps / x) * 0.5;
	} else {
		tmp = x - (x + (-0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-7.5d-195)) .or. (.not. (eps <= 1.15d-164))) then
        tmp = (eps / x) * 0.5d0
    else
        tmp = x - (x + ((-0.5d0) * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.5e-195) || !(eps <= 1.15e-164)) {
		tmp = (eps / x) * 0.5;
	} else {
		tmp = x - (x + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -7.5e-195) or not (eps <= 1.15e-164):
		tmp = (eps / x) * 0.5
	else:
		tmp = x - (x + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -7.5e-195) || !(eps <= 1.15e-164))
		tmp = Float64(Float64(eps / x) * 0.5);
	else
		tmp = Float64(x - Float64(x + Float64(-0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -7.5e-195) || ~((eps <= 1.15e-164)))
		tmp = (eps / x) * 0.5;
	else
		tmp = x - (x + (-0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -7.5e-195], N[Not[LessEqual[eps, 1.15e-164]], $MachinePrecision]], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision], N[(x - N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.5e-195 or 1.14999999999999993e-164 < eps

   1. Initial program 76.6%

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

   3. Taylor expanded in x around inf 29.8%

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

   if -7.5e-195 < eps < 1.14999999999999993e-164

   1. Initial program 81.8%

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

   3. Taylor expanded in x around inf 45.0%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-195} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-216} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.6e-216) (not (<= eps 3.6e-165)))
   (* (/ eps x) 0.5)
   (- x x)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-216) || !(eps <= 3.6e-165)) {
		tmp = (eps / x) * 0.5;
	} else {
		tmp = x - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.6d-216)) .or. (.not. (eps <= 3.6d-165))) then
        tmp = (eps / x) * 0.5d0
    else
        tmp = x - x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-216) || !(eps <= 3.6e-165)) {
		tmp = (eps / x) * 0.5;
	} else {
		tmp = x - x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.6e-216) or not (eps <= 3.6e-165):
		tmp = (eps / x) * 0.5
	else:
		tmp = x - x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.6e-216) || !(eps <= 3.6e-165))
		tmp = Float64(Float64(eps / x) * 0.5);
	else
		tmp = Float64(x - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.6e-216) || ~((eps <= 3.6e-165)))
		tmp = (eps / x) * 0.5;
	else
		tmp = x - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.6e-216], N[Not[LessEqual[eps, 3.6e-165]], $MachinePrecision]], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision], N[(x - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.59999999999999993e-216 or 3.59999999999999984e-165 < eps

   1. Initial program 78.5%

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

   3. Taylor expanded in x around inf 27.9%

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

   if -4.59999999999999993e-216 < eps < 3.59999999999999984e-165

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf 46.9%

      \[\leadsto x - \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-216} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 5.2% 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 78.9%

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

   1. flip-- 78.8%

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

   2. div-inv 78.6%

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

   3. add-sqr-sqrt 78.7%

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

   4. associate--r- 80.5%

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

   5. pow2 80.4%

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

   6. pow2 80.5%

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

   7. sub-neg 80.5%

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

   8. add-sqr-sqrt 67.6%

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

   9. hypot-def 67.6%

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

4. Applied egg-rr 67.6%

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

   1. +-inverses 67.6%

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

   2. +-lft-identity 67.6%

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

   3. associate-*r/ 67.7%

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

   4. associate-/l* 67.7%

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

   5. /-rgt-identity 67.7%

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

6. Simplified 67.7%

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

7. Taylor expanded in x around inf 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-def 0.0%

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

   4. associate-*r/ 0.0%

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

   5. *-commutative 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 29.1%

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

   8. associate-*r* 29.1%

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

   9. metadata-eval 29.1%

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

   10. associate-*r/ 29.1%

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

   11. *-commutative 29.1%

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

9. Simplified 29.1%

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

10. Taylor expanded in eps around inf 5.6%

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

    1. *-commutative 5.6%

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

12. Simplified 5.6%

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

13. Final simplification 5.6%

    \[\leadsto x \cdot -2 \]
14. Add Preprocessing

Alternative 10: 21.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in x around inf 23.4%

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

4. Final simplification 23.4%

   \[\leadsto x - x \]
5. Add Preprocessing

Alternative 11: 3.5% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x

Julia

function code(x, eps)
	return x
end

MATLAB

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

Wolfram

code[x_, eps_] := x

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in x around 0 52.9%

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

   1. neg-mul-1 52.9%

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

5. Simplified 52.9%

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

6. Taylor expanded in x around inf 3.4%

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

7. Final simplification 3.4%

   \[\leadsto x \]
8. Add Preprocessing

Developer target: 82.2% 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 2024031 
(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)))

  :herbie-target
  (/ eps (+ x (sqrt (- (* x x) eps))))

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