ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.8% → 99.2%

Time: 10.0s

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\varepsilon \cdot \frac{-0.5}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (+ (* eps (/ -0.5 x)) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-154:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(Float64(eps * Float64(-0.5 / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / ((eps * (-0.5 / x)) + (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], -2e-154], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 98.5%

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

      1. flip-- 98.5%

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

      2. div-inv 98.1%

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

      3. add-sqr-sqrt 98.0%

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

      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 99.3%

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

      9. hypot-define 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. +-inverses 99.3%

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

      3. +-lft-identity 99.3%

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

      4. associate-*l/ 99.3%

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

      5. *-lft-identity 99.3%

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

   6. Simplified 99.3%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.1%

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

      1. flip-- 8.1%

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

      2. div-inv 8.1%

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

      3. add-sqr-sqrt 8.4%

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

      4. associate--r- 99.7%

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

      5. pow2 98.9%

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

      6. pow2 99.7%

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

      7. sub-neg 99.7%

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

      8. add-sqr-sqrt 48.0%

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

      9. hypot-define 48.0%

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

   4. Applied egg-rr 48.0%

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

      1. *-commutative 48.0%

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

      2. +-inverses 48.0%

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

      3. +-lft-identity 48.0%

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

      4. associate-*l/ 48.1%

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

      5. *-lft-identity 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 98.8%

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

      4. neg-mul-1 98.8%

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

      5. distribute-neg-frac 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. distribute-lft-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

      9. associate-*r/ 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-/l* 98.8%

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

      12. *-commutative 98.8%

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

   9. Simplified 98.8%

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

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\varepsilon \cdot \frac{-0.5}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (hypot (sqrt (- eps)) x))
   (/ eps (+ (* eps (/ -0.5 x)) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - Math.hypot(Math.sqrt(-eps), x);
	} else {
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-154:
		tmp = x - math.hypot(math.sqrt(-eps), x)
	else:
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / Float64(Float64(eps * Float64(-0.5 / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154)
		tmp = x - hypot(sqrt(-eps), x);
	else
		tmp = eps / ((eps * (-0.5 / x)) + (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], -2e-154], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 98.5%

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

      1. sub-neg 98.5%

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

      2. +-commutative 98.5%

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

      3. add-sqr-sqrt 98.5%

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

      4. hypot-define 98.5%

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

   4. Applied egg-rr 98.5%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.1%

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

      1. flip-- 8.1%

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

      2. div-inv 8.1%

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

      3. add-sqr-sqrt 8.4%

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

      4. associate--r- 99.7%

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

      5. pow2 98.9%

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

      6. pow2 99.7%

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

      7. sub-neg 99.7%

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

      8. add-sqr-sqrt 48.0%

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

      9. hypot-define 48.0%

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

   4. Applied egg-rr 48.0%

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

      1. *-commutative 48.0%

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

      2. +-inverses 48.0%

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

      3. +-lft-identity 48.0%

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

      4. associate-*l/ 48.1%

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

      5. *-lft-identity 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 98.8%

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

      4. neg-mul-1 98.8%

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

      5. distribute-neg-frac 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. distribute-lft-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

      9. associate-*r/ 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-/l* 98.8%

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

      12. *-commutative 98.8%

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

   9. Simplified 98.8%

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

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\varepsilon \cdot \frac{-0.5}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154) t_0 (/ eps (+ (* eps (/ -0.5 x)) (* x 2.0))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = eps / ((eps * (-0.5 / 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(((x * x) - eps))
    if (t_0 <= (-2d-154)) then
        tmp = t_0
    else
        tmp = eps / ((eps * ((-0.5d0) / 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(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-154:
		tmp = t_0
	else:
		tmp = eps / ((eps * (-0.5 / x)) + (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 <= -2e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(eps * Float64(-0.5 / x)) + 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 <= -2e-154)
		tmp = t_0;
	else
		tmp = eps / ((eps * (-0.5 / x)) + (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, -2e-154], t$95$0, N[(eps / N[(N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 98.5%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.1%

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

      1. flip-- 8.1%

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

      2. div-inv 8.1%

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

      3. add-sqr-sqrt 8.4%

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

      4. associate--r- 99.7%

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

      5. pow2 98.9%

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

      6. pow2 99.7%

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

      7. sub-neg 99.7%

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

      8. add-sqr-sqrt 48.0%

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

      9. hypot-define 48.0%

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

   4. Applied egg-rr 48.0%

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

      1. *-commutative 48.0%

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

      2. +-inverses 48.0%

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

      3. +-lft-identity 48.0%

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

      4. associate-*l/ 48.1%

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

      5. *-lft-identity 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 98.8%

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

      4. neg-mul-1 98.8%

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

      5. distribute-neg-frac 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. distribute-lft-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

      9. associate-*r/ 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-/l* 98.8%

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

      12. *-commutative 98.8%

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

   9. Simplified 98.8%

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

Alternative 4: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-121}:\\ \;\;\;\;-\sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\varepsilon \cdot \frac{-0.5}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.8e-121)
   (- (sqrt (- eps)))
   (/ eps (+ (* eps (/ -0.5 x)) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.8e-121) {
		tmp = -sqrt(-eps);
	} else {
		tmp = eps / ((eps * (-0.5 / 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) :: tmp
    if (x <= 2.8d-121) then
        tmp = -sqrt(-eps)
    else
        tmp = eps / ((eps * ((-0.5d0) / x)) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.8e-121) {
		tmp = -Math.sqrt(-eps);
	} else {
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.8e-121:
		tmp = -math.sqrt(-eps)
	else:
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.8e-121)
		tmp = Float64(-sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(eps * Float64(-0.5 / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.8e-121)
		tmp = -sqrt(-eps);
	else
		tmp = eps / ((eps * (-0.5 / x)) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.8e-121], (-N[Sqrt[(-eps)], $MachinePrecision]), N[(eps / N[(N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e-121

   1. Initial program 96.9%

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

      1. flip-- 96.8%

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

      2. div-inv 96.4%

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

      3. add-sqr-sqrt 96.2%

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

      4. associate--r- 99.3%

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

      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 96.8%

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

      9. hypot-define 96.8%

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

   4. Applied egg-rr 96.8%

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

      1. *-commutative 96.8%

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

      2. +-inverses 96.8%

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

      3. +-lft-identity 96.8%

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

      4. associate-*l/ 96.9%

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

      5. *-lft-identity 96.9%

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

   6. Simplified 96.9%

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

      1. add-cube-cbrt 96.0%

         \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \sqrt[3]{\varepsilon}}}\right)} \]

      2. distribute-rgt-neg-in 96.0%

         \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

      3. pow2 96.0%

         \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2}} \cdot \left(-\sqrt[3]{\varepsilon}\right)}\right)} \]

   8. Applied egg-rr 96.0%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2} \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

   9. Taylor expanded in eps around -inf 96.8%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{\varepsilon}{{\left(\sqrt[3]{-1}\right)}^{3}}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 96.8%

          \[\leadsto \color{blue}{-\sqrt{\frac{\varepsilon}{{\left(\sqrt[3]{-1}\right)}^{3}}}} \]

       2. rem-cube-cbrt 96.8%

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

       3. metadata-eval 96.8%

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

       4. distribute-neg-frac2 96.8%

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

       5. /-rgt-identity 96.8%

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

   11. Simplified 96.8%

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

   if 2.8000000000000001e-121 < x

   1. Initial program 27.1%

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

      1. flip-- 27.2%

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

      2. div-inv 27.1%

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

      3. add-sqr-sqrt 27.4%

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

      4. associate--r- 99.7%

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

      5. pow2 99.0%

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

      6. pow2 99.7%

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

      7. sub-neg 99.7%

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

      8. add-sqr-sqrt 60.2%

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

      9. hypot-define 60.2%

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

   4. Applied egg-rr 60.2%

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

      1. *-commutative 60.2%

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

      2. +-inverses 60.2%

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

      3. +-lft-identity 60.2%

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

      4. associate-*l/ 60.3%

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

      5. *-lft-identity 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 82.3%

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

      4. neg-mul-1 82.3%

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

      5. distribute-neg-frac 82.3%

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

      6. distribute-rgt-neg-in 82.3%

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

      7. distribute-lft-neg-in 82.3%

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

      8. metadata-eval 82.3%

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

      9. associate-*r/ 82.3%

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

      10. *-commutative 82.3%

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

      11. associate-/l* 82.3%

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

      12. *-commutative 82.3%

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

   9. Simplified 82.3%

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

Alternative 5: 45.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. flip-- 60.4%

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

   2. div-inv 60.1%

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

   3. add-sqr-sqrt 60.2%

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

   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 99.1%

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

   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 77.7%

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

   9. hypot-define 77.7%

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

4. Applied egg-rr 77.7%

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

   1. *-commutative 77.7%

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

   2. +-inverses 77.7%

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

   3. +-lft-identity 77.7%

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

   4. associate-*l/ 77.7%

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

   5. *-lft-identity 77.7%

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

6. Simplified 77.7%

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

7. Taylor expanded in eps around 0 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 47.0%

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

   4. neg-mul-1 47.0%

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

   5. distribute-neg-frac 47.0%

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

   6. distribute-rgt-neg-in 47.0%

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

   7. distribute-lft-neg-in 47.0%

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

   8. metadata-eval 47.0%

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

   9. associate-*r/ 47.0%

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

   10. *-commutative 47.0%

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

   11. associate-/l* 47.0%

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

   12. *-commutative 47.0%

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

9. Simplified 47.0%

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

Alternative 6: 45.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. flip-- 60.4%

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

   2. div-inv 60.1%

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

   3. add-sqr-sqrt 60.2%

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

   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 99.1%

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

   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 77.7%

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

   9. hypot-define 77.7%

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

4. Applied egg-rr 77.7%

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

   1. *-commutative 77.7%

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

   2. +-inverses 77.7%

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

   3. +-lft-identity 77.7%

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

   4. associate-*l/ 77.7%

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

   5. *-lft-identity 77.7%

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

6. Simplified 77.7%

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

   1. add-cube-cbrt 77.2%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \sqrt[3]{\varepsilon}}}\right)} \]

   2. distribute-rgt-neg-in 77.2%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

   3. pow2 77.2%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2}} \cdot \left(-\sqrt[3]{\varepsilon}\right)}\right)} \]

8. Applied egg-rr 77.2%

   \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2} \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

9. Taylor expanded in eps around 0 0.0%

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

    1. associate-*r/ 0.0%

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

    2. *-commutative 0.0%

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

    3. unpow2 0.0%

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

    4. rem-square-sqrt 47.0%

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

    5. neg-mul-1 47.0%

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

    6. distribute-rgt-neg-in 47.0%

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

    7. distribute-lft-neg-in 47.0%

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

    8. metadata-eval 47.0%

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

    9. associate-*r/ 47.0%

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

    10. *-commutative 47.0%

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

11. Simplified 47.0%

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

12. Final simplification 47.0%

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

Alternative 7: 45.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. flip-- 60.4%

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

   2. clear-num 60.1%

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

   3. sub-neg 60.1%

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

   4. add-sqr-sqrt 57.8%

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

   5. hypot-define 57.8%

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

   6. add-sqr-sqrt 57.7%

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

   7. associate--r- 77.3%

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

   8. pow2 77.0%

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

   9. pow2 77.3%

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

4. Applied egg-rr 77.3%

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

5. Taylor expanded in eps around 0 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 46.5%

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

   4. neg-mul-1 46.5%

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

   5. distribute-neg-frac 46.5%

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

   6. distribute-rgt-neg-in 46.5%

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

   7. distribute-lft-neg-in 46.5%

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

   8. metadata-eval 46.5%

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

   9. associate-*r/ 46.5%

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

   10. *-commutative 46.5%

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

   11. associate-/l* 46.5%

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

   12. *-commutative 46.5%

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

7. Simplified 46.5%

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

8. Taylor expanded in eps around inf 46.5%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{x}}} \]
9. Step-by-step derivation

   1. cancel-sign-sub-inv 46.5%

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

   2. metadata-eval 46.5%

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

   3. div-inv 46.5%

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

10. Applied egg-rr 46.5%

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

11. Final simplification 46.5%

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

Alternative 8: 44.6% accurate, 21.4× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / (x + x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. flip-- 60.4%

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

   2. div-inv 60.1%

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

   3. add-sqr-sqrt 60.2%

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

   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 99.1%

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

   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 77.7%

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

   9. hypot-define 77.7%

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

4. Applied egg-rr 77.7%

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

   1. *-commutative 77.7%

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

   2. +-inverses 77.7%

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

   3. +-lft-identity 77.7%

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

   4. associate-*l/ 77.7%

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

   5. *-lft-identity 77.7%

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

6. Simplified 77.7%

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

7. Taylor expanded in x around inf 45.9%

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

Alternative 9: 44.6% 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 60.4%

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

3. Taylor expanded in x around inf 45.9%

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

4. Final simplification 45.9%

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

Alternative 10: 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 60.4%

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

   1. flip-- 60.4%

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

   2. clear-num 60.1%

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

   3. sub-neg 60.1%

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

   4. add-sqr-sqrt 57.8%

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

   5. hypot-define 57.8%

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

   6. add-sqr-sqrt 57.7%

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

   7. associate--r- 77.3%

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

   8. pow2 77.0%

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

   9. pow2 77.3%

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

4. Applied egg-rr 77.3%

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

5. Taylor expanded in eps around 0 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 46.5%

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

   4. neg-mul-1 46.5%

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

   5. distribute-neg-frac 46.5%

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

   6. distribute-rgt-neg-in 46.5%

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

   7. distribute-lft-neg-in 46.5%

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

   8. metadata-eval 46.5%

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

   9. associate-*r/ 46.5%

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

   10. *-commutative 46.5%

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

   11. associate-/l* 46.5%

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

   12. *-commutative 46.5%

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

7. Simplified 46.5%

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

8. Taylor expanded in eps around inf 5.2%

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

   1. *-commutative 5.2%

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

10. Simplified 5.2%

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

Alternative 11: 4.3% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in x around inf 4.3%

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

4. Taylor expanded in x around 0 4.3%

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

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024113 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4d"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))

  :alt
  (! :herbie-platform default (/ eps (+ x (sqrt (- (* x x) eps)))))

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