ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.6% → 99.3%

Time: 9.2s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.6% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \varepsilon \cdot \frac{-0.5}{x}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-154)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/
    eps
    (+ x (+ x (+ (/ -0.125 (/ x (pow (/ eps x) 2.0))) (* eps (/ -0.5 x))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / (x + (x + ((-0.125 / (x / pow((eps / x), 2.0))) + (eps * (-0.5 / x)))));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-154) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / (x + (x + ((-0.125 / (x / Math.pow((eps / x), 2.0))) + (eps * (-0.5 / x)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-154:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / (x + (x + ((-0.125 / (x / math.pow((eps / x), 2.0))) + (eps * (-0.5 / x)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-154)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(-0.125 / Float64(x / (Float64(eps / x) ^ 2.0))) + Float64(eps * Float64(-0.5 / x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / (x + (x + ((-0.125 / (x / ((eps / x) ^ 2.0))) + (eps * (-0.5 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-154], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(N[(-0.125 / N[(x / N[Power[N[(eps / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.8%

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

      1. flip-- 98.7%

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

      2. div-inv 98.5%

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

      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- 99.1%

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

      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 99.1%

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

      7. sub-neg 99.1%

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

      8. add-sqr-sqrt 99.1%

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

      9. hypot-def 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. +-inverses 99.1%

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

      2. +-lft-identity 99.1%

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

      3. associate-*r/ 99.2%

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

      4. associate-/l* 99.2%

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

      5. /-rgt-identity 99.2%

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

   6. Simplified 99.2%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.0%

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

      1. flip-- 9.1%

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

      2. div-inv 9.1%

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

      3. add-sqr-sqrt 9.2%

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

      4. associate--r- 99.6%

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

      5. pow2 98.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 42.4%

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

      9. hypot-def 42.4%

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

   4. Applied egg-rr 42.4%

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

      1. +-inverses 42.4%

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

      2. +-lft-identity 42.4%

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

      3. associate-*r/ 42.6%

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

      4. associate-/l* 42.6%

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

      5. /-rgt-identity 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \left(-0.125 \cdot \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{4}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)}} \]
   8. Step-by-step derivation

      1. fma-def 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{4}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}\right)} \]

      2. metadata-eval 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{\color{blue}{\left(2 \cdot 2\right)}}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      3. pow-sqr 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      4. unpow2 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      5. rem-square-sqrt 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(\color{blue}{-1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      7. rem-square-sqrt 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(-1 \cdot \color{blue}{-1}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      8. metadata-eval 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \color{blue}{1}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      9. *-rgt-identity 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\color{blue}{{\varepsilon}^{2}}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      10. associate-*r/ 0.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \color{blue}{\frac{0.5 \cdot \left(\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}}\right)\right)} \]

      11. *-commutative 0.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon\right)}}{x}\right)\right)} \]

      12. unpow2 0.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \varepsilon\right)}{x}\right)\right)} \]

      13. rem-square-sqrt 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{0.5 \cdot \left(\color{blue}{-1} \cdot \varepsilon\right)}{x}\right)\right)} \]

      14. associate-*r* 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot \varepsilon}}{x}\right)\right)} \]

      15. metadata-eval 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{\color{blue}{-0.5} \cdot \varepsilon}{x}\right)\right)} \]

      16. associate-*r/ 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \color{blue}{-0.5 \cdot \frac{\varepsilon}{x}}\right)\right)} \]

      17. *-commutative 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \color{blue}{\frac{\varepsilon}{x} \cdot -0.5}\right)\right)} \]

   9. Simplified 90.4%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\color{blue}{1 \cdot {\varepsilon}^{2}}}{{x}^{3}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       2. cube-mult 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1 \cdot {\varepsilon}^{2}}{\color{blue}{x \cdot \left(x \cdot x\right)}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       3. times-frac 99.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{1}{x} \cdot \frac{{\varepsilon}^{2}}{x \cdot x}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       4. unpow2 99.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1}{x} \cdot \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{x \cdot x}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       5. frac-times 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1}{x} \cdot \color{blue}{\left(\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}\right)}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       6. pow2 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1}{x} \cdot \color{blue}{{\left(\frac{\varepsilon}{x}\right)}^{2}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{1}{x} \cdot {\left(\frac{\varepsilon}{x}\right)}^{2}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{1 \cdot {\left(\frac{\varepsilon}{x}\right)}^{2}}{x}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       2. *-lft-identity 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\color{blue}{{\left(\frac{\varepsilon}{x}\right)}^{2}}}{x}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{{\left(\frac{\varepsilon}{x}\right)}^{2}}{x}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]
   14. Step-by-step derivation

       1. fma-udef 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \color{blue}{\left(-0.125 \cdot \frac{{\left(\frac{\varepsilon}{x}\right)}^{2}}{x} + \frac{\varepsilon}{x} \cdot -0.5\right)}\right)} \]

       2. clear-num 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(-0.125 \cdot \color{blue}{\frac{1}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}}} + \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       3. un-div-inv 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(\color{blue}{\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}}} + \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       4. associate-*l/ 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \color{blue}{\frac{\varepsilon \cdot -0.5}{x}}\right)\right)} \]

       5. associate-*r/ 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \color{blue}{\varepsilon \cdot \frac{-0.5}{x}}\right)\right)} \]

   15. Applied egg-rr 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \color{blue}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \varepsilon \cdot \frac{-0.5}{x}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \varepsilon \cdot \frac{-0.5}{x}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \varepsilon \cdot \frac{-0.5}{x}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154)
     t_0
     (/
      eps
      (+
       x
       (+ x (+ (/ -0.125 (/ x (pow (/ eps x) 2.0))) (* eps (/ -0.5 x)))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + ((-0.125 / (x / pow((eps / x), 2.0))) + (eps * (-0.5 / x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-154)) then
        tmp = t_0
    else
        tmp = eps / (x + (x + (((-0.125d0) / (x / ((eps / x) ** 2.0d0))) + (eps * ((-0.5d0) / x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + ((-0.125 / (x / Math.pow((eps / x), 2.0))) + (eps * (-0.5 / x)))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-154:
		tmp = t_0
	else:
		tmp = eps / (x + (x + ((-0.125 / (x / math.pow((eps / x), 2.0))) + (eps * (-0.5 / x)))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(-0.125 / Float64(x / (Float64(eps / x) ^ 2.0))) + Float64(eps * Float64(-0.5 / x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = eps / (x + (x + ((-0.125 / (x / ((eps / x) ^ 2.0))) + (eps * (-0.5 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-154], t$95$0, N[(eps / N[(x + N[(x + N[(N[(-0.125 / N[(x / N[Power[N[(eps / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.8%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.0%

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

      1. flip-- 9.1%

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

      2. div-inv 9.1%

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

      3. add-sqr-sqrt 9.2%

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

      4. associate--r- 99.6%

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

      5. pow2 98.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 42.4%

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

      9. hypot-def 42.4%

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

   4. Applied egg-rr 42.4%

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

      1. +-inverses 42.4%

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

      2. +-lft-identity 42.4%

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

      3. associate-*r/ 42.6%

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

      4. associate-/l* 42.6%

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

      5. /-rgt-identity 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \left(-0.125 \cdot \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{4}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)}} \]
   8. Step-by-step derivation

      1. fma-def 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{4}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}\right)} \]

      2. metadata-eval 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{\color{blue}{\left(2 \cdot 2\right)}}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      3. pow-sqr 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      4. unpow2 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      5. rem-square-sqrt 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(\color{blue}{-1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      7. rem-square-sqrt 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \left(-1 \cdot \color{blue}{-1}\right)}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      8. metadata-eval 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2} \cdot \color{blue}{1}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      9. *-rgt-identity 0.0%

         \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\color{blue}{{\varepsilon}^{2}}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)} \]

      10. associate-*r/ 0.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \color{blue}{\frac{0.5 \cdot \left(\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}}\right)\right)} \]

      11. *-commutative 0.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon\right)}}{x}\right)\right)} \]

      12. unpow2 0.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \varepsilon\right)}{x}\right)\right)} \]

      13. rem-square-sqrt 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{0.5 \cdot \left(\color{blue}{-1} \cdot \varepsilon\right)}{x}\right)\right)} \]

      14. associate-*r* 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot \varepsilon}}{x}\right)\right)} \]

      15. metadata-eval 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{\color{blue}{-0.5} \cdot \varepsilon}{x}\right)\right)} \]

      16. associate-*r/ 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \color{blue}{-0.5 \cdot \frac{\varepsilon}{x}}\right)\right)} \]

      17. *-commutative 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \color{blue}{\frac{\varepsilon}{x} \cdot -0.5}\right)\right)} \]

   9. Simplified 90.4%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \mathsf{fma}\left(-0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\color{blue}{1 \cdot {\varepsilon}^{2}}}{{x}^{3}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       2. cube-mult 90.4%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1 \cdot {\varepsilon}^{2}}{\color{blue}{x \cdot \left(x \cdot x\right)}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       3. times-frac 99.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{1}{x} \cdot \frac{{\varepsilon}^{2}}{x \cdot x}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       4. unpow2 99.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1}{x} \cdot \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{x \cdot x}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       5. frac-times 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1}{x} \cdot \color{blue}{\left(\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}\right)}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       6. pow2 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{1}{x} \cdot \color{blue}{{\left(\frac{\varepsilon}{x}\right)}^{2}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{1}{x} \cdot {\left(\frac{\varepsilon}{x}\right)}^{2}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{1 \cdot {\left(\frac{\varepsilon}{x}\right)}^{2}}{x}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       2. *-lft-identity 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\color{blue}{{\left(\frac{\varepsilon}{x}\right)}^{2}}}{x}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{{\left(\frac{\varepsilon}{x}\right)}^{2}}{x}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]
   14. Step-by-step derivation

       1. fma-udef 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \color{blue}{\left(-0.125 \cdot \frac{{\left(\frac{\varepsilon}{x}\right)}^{2}}{x} + \frac{\varepsilon}{x} \cdot -0.5\right)}\right)} \]

       2. clear-num 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(-0.125 \cdot \color{blue}{\frac{1}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}}} + \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       3. un-div-inv 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(\color{blue}{\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}}} + \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       4. associate-*l/ 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \color{blue}{\frac{\varepsilon \cdot -0.5}{x}}\right)\right)} \]

       5. associate-*r/ 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \color{blue}{\varepsilon \cdot \frac{-0.5}{x}}\right)\right)} \]

   15. Applied egg-rr 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \color{blue}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \varepsilon \cdot \frac{-0.5}{x}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \varepsilon \cdot \frac{-0.5}{x}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154) t_0 (/ eps (+ x (+ x (* (/ eps x) -0.5)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-154)) then
        tmp = t_0
    else
        tmp = eps / (x + (x + ((eps / x) * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-154:
		tmp = t_0
	else:
		tmp = eps / (x + (x + ((eps / x) * -0.5)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(eps / x) * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-154], t$95$0, N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.8%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.0%

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

      1. flip-- 9.1%

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

      2. div-inv 9.1%

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

      3. add-sqr-sqrt 9.2%

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

      4. associate--r- 99.6%

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

      5. pow2 98.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 42.4%

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

      9. hypot-def 42.4%

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

   4. Applied egg-rr 42.4%

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

      1. +-inverses 42.4%

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

      2. +-lft-identity 42.4%

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

      3. associate-*r/ 42.6%

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

      4. associate-/l* 42.6%

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

      5. /-rgt-identity 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 99.5%

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

      6. neg-mul-1 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in eps around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-104}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 7.5e-104)
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (* (/ eps x) -0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7.5e-104) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.5d-104) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + ((eps / x) * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.5e-104) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 7.5e-104:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + ((eps / x) * -0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7.5e-104)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(eps / x) * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.5e-104)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7.5e-104], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e-104

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0 92.9%

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

      1. neg-mul-1 92.9%

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

   5. Simplified 92.9%

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

   if 7.5e-104 < x

   1. Initial program 28.9%

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

      1. flip-- 28.9%

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

      2. div-inv 28.9%

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

      3. add-sqr-sqrt 28.9%

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

      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 98.6%

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

      6. pow2 99.5%

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

      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 56.8%

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

      9. hypot-def 56.8%

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

   4. Applied egg-rr 56.8%

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

      1. +-inverses 56.8%

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

      2. +-lft-identity 56.8%

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

      3. associate-*r/ 56.9%

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

      4. associate-/l* 56.9%

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

      5. /-rgt-identity 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 80.1%

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

      6. neg-mul-1 80.1%

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

   9. Simplified 80.1%

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

   10. Taylor expanded in eps around 0 80.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-104}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

   1. flip-- 65.8%

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

   2. div-inv 65.6%

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

   3. add-sqr-sqrt 65.4%

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

   4. associate--r- 99.3%

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

   5. pow2 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 99.3%

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

   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 78.3%

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

   9. hypot-def 78.3%

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

4. Applied egg-rr 78.3%

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

   1. +-inverses 78.3%

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

   2. +-lft-identity 78.3%

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

   3. associate-*r/ 78.4%

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

   4. associate-/l* 78.4%

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

   5. /-rgt-identity 78.4%

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

6. Simplified 78.4%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 41.8%

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

   6. neg-mul-1 41.8%

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

9. Simplified 41.8%

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

10. Taylor expanded in eps around 0 41.8%

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

11. Final simplification 41.8%

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

Alternative 6: 43.8% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in x around inf 40.6%

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

4. Final simplification 40.6%

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

Alternative 7: 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 65.9%

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

   1. flip-- 65.8%

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

   2. div-inv 65.6%

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

   3. add-sqr-sqrt 65.4%

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

   4. associate--r- 99.3%

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

   5. pow2 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 99.3%

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

   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 78.3%

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

   9. hypot-def 78.3%

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

4. Applied egg-rr 78.3%

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

   1. +-inverses 78.3%

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

   2. +-lft-identity 78.3%

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

   3. associate-*r/ 78.4%

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

   4. associate-/l* 78.4%

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

   5. /-rgt-identity 78.4%

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

6. Simplified 78.4%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 41.8%

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

   6. neg-mul-1 41.8%

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

9. Simplified 41.8%

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

10. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

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

12. Simplified 5.3%

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

13. Final simplification 5.3%

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

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024036 
(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))))