ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.2% → 99.3%

Time: 11.6s

Alternatives: 12

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.125, \frac{\frac{\varepsilon}{{x}^{2}}}{x}, \mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5e-310)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/
    1.0
    (fma -0.125 (/ (/ eps (pow x 2.0)) x) (fma 2.0 (/ x eps) (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5e-310) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = 1.0 / fma(-0.125, ((eps / pow(x, 2.0)) / x), fma(2.0, (x / eps), (-0.5 / x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -5e-310)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(1.0 / fma(-0.125, Float64(Float64(eps / (x ^ 2.0)) / x), fma(2.0, Float64(x / eps), Float64(-0.5 / x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5e-310], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-0.125 * N[(N[(eps / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(x / eps), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.999999999999985e-310

   1. Initial program 73.4%

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

      1. flip-- 73.4%

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

      2. div-inv 73.2%

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

      3. add-sqr-sqrt 73.0%

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

      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 99.4%

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

      9. hypot-def 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. +-inverses 99.4%

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

      2. +-lft-identity 99.4%

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

      3. associate-*r/ 99.4%

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

      4. associate-/l* 99.4%

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

      5. /-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if -4.999999999999985e-310 < eps

   1. Initial program 12.6%

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

      1. flip-- 12.7%

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

      2. div-inv 12.7%

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

      3. add-sqr-sqrt 12.7%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r/ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. metadata-eval 0.0%

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

      3. pow-sqr 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 0.0%

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

      8. metadata-eval 0.0%

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

      9. *-rgt-identity 0.0%

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

      10. +-commutative 0.0%

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

      11. fma-def 0.0%

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

      12. associate-*r/ 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 93.2%

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

      15. metadata-eval 93.2%

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

   9. Simplified 93.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{3}}, \mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)\right)}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 93.2%

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

       2. cube-mult 93.2%

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

       3. unpow2 93.2%

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

       4. times-frac 96.7%

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

   11. Applied egg-rr 96.7%

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

       1. associate-*l/ 96.7%

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

       2. *-lft-identity 96.7%

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

   13. Simplified 96.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-0.125, \color{blue}{\frac{\frac{\varepsilon}{{x}^{2}}}{x}}, \mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.125, \frac{\frac{\varepsilon}{{x}^{2}}}{x}, \mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-151)
   (/ 1.0 (/ (+ x (hypot x (sqrt (- eps)))) eps))
   (/ eps (+ (* x 2.0) (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-151) {
		tmp = 1.0 / ((x + hypot(x, sqrt(-eps))) / eps);
	} else {
		tmp = 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))) <= -2e-151) {
		tmp = 1.0 / ((x + Math.hypot(x, Math.sqrt(-eps))) / eps);
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-151:
		tmp = 1.0 / ((x + math.hypot(x, math.sqrt(-eps))) / eps)
	else:
		tmp = 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))) <= -2e-151)
		tmp = Float64(1.0 / Float64(Float64(x + hypot(x, sqrt(Float64(-eps)))) / eps));
	else
		tmp = Float64(eps / Float64(Float64(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))) <= -2e-151)
		tmp = 1.0 / ((x + hypot(x, sqrt(-eps))) / eps);
	else
		tmp = 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], -2e-151], N[(1.0 / N[(N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-151

   1. Initial program 98.3%

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

      1. flip-- 98.2%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 97.7%

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

      4. associate--r- 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. sub-neg 99.2%

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

      8. add-sqr-sqrt 99.3%

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

      9. hypot-def 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. associate-/r/ 99.3%

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

      3. +-inverses 99.3%

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

      4. +-commutative 99.3%

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

   6. Simplified 99.3%

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

   if -1.9999999999999999e-151 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.5%

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

      1. flip-- 9.5%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 50.7%

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

      9. hypot-def 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. +-inverses 50.7%

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

      2. +-lft-identity 50.7%

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

      3. associate-*r/ 50.9%

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

      4. associate-/l* 50.9%

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

      5. /-rgt-identity 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.0%

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

      8. associate-*r* 98.0%

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

      9. metadata-eval 98.0%

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

      10. associate-*r/ 98.0%

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

      11. *-commutative 98.0%

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

   9. Simplified 98.0%

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

       1. fma-udef 98.0%

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

       2. associate-*l/ 98.0%

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

       3. associate-/l* 98.0%

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

       4. div-inv 98.0%

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

       5. clear-num 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-151)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (+ (* x 2.0) (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-151) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = 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))) <= -2e-151) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-151:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = 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))) <= -2e-151)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(Float64(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))) <= -2e-151)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = 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], -2e-151], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-151

   1. Initial program 98.3%

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

      1. flip-- 98.2%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 97.7%

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

      4. associate--r- 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. sub-neg 99.2%

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

      8. add-sqr-sqrt 99.3%

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

      9. hypot-def 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. +-inverses 99.3%

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

      2. +-lft-identity 99.3%

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

      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 -1.9999999999999999e-151 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.5%

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

      1. flip-- 9.5%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 50.7%

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

      9. hypot-def 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. +-inverses 50.7%

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

      2. +-lft-identity 50.7%

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

      3. associate-*r/ 50.9%

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

      4. associate-/l* 50.9%

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

      5. /-rgt-identity 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.0%

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

      8. associate-*r* 98.0%

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

      9. metadata-eval 98.0%

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

      10. associate-*r/ 98.0%

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

      11. *-commutative 98.0%

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

   9. Simplified 98.0%

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

       1. fma-udef 98.0%

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

       2. associate-*l/ 98.0%

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

       3. associate-/l* 98.0%

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

       4. div-inv 98.0%

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

       5. clear-num 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-151)
   (- x (sqrt (fma x x (- eps))))
   (/ eps (+ (* x 2.0) (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-151) {
		tmp = x - sqrt(fma(x, x, -eps));
	} else {
		tmp = 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))) <= -2e-151)
		tmp = Float64(x - sqrt(fma(x, x, Float64(-eps))));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(eps * Float64(-0.5 / x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-151], N[(x - N[Sqrt[N[(x * x + (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-151

   1. Initial program 98.3%

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

      1. fma-neg 98.3%

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

   4. Applied egg-rr 98.3%

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

   if -1.9999999999999999e-151 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.5%

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

      1. flip-- 9.5%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 50.7%

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

      9. hypot-def 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. +-inverses 50.7%

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

      2. +-lft-identity 50.7%

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

      3. associate-*r/ 50.9%

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

      4. associate-/l* 50.9%

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

      5. /-rgt-identity 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.0%

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

      8. associate-*r* 98.0%

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

      9. metadata-eval 98.0%

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

      10. associate-*r/ 98.0%

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

      11. *-commutative 98.0%

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

   9. Simplified 98.0%

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

       1. fma-udef 98.0%

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

       2. associate-*l/ 98.0%

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

       3. associate-/l* 98.0%

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

       4. div-inv 98.0%

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

       5. clear-num 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-151) t_0 (/ 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 <= -2e-151) {
		tmp = t_0;
	} else {
		tmp = 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 <= (-2d-151)) then
        tmp = t_0
    else
        tmp = 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 <= -2e-151) {
		tmp = t_0;
	} else {
		tmp = 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 <= -2e-151:
		tmp = t_0
	else:
		tmp = 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 <= -2e-151)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(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 <= -2e-151)
		tmp = t_0;
	else
		tmp = 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, -2e-151], t$95$0, N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-151

   1. Initial program 98.3%

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

   if -1.9999999999999999e-151 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.5%

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

      1. flip-- 9.5%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 50.7%

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

      9. hypot-def 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. +-inverses 50.7%

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

      2. +-lft-identity 50.7%

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

      3. associate-*r/ 50.9%

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

      4. associate-/l* 50.9%

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

      5. /-rgt-identity 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.0%

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

      8. associate-*r* 98.0%

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

      9. metadata-eval 98.0%

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

      10. associate-*r/ 98.0%

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

      11. *-commutative 98.0%

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

   9. Simplified 98.0%

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

       1. fma-udef 98.0%

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

       2. associate-*l/ 98.0%

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

       3. associate-/l* 98.0%

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

       4. div-inv 98.0%

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

       5. clear-num 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.05 \cdot 10^{-109} \lor \neg \left(x \leq 1.02 \cdot 10^{-102}\right) \land x \leq 3.2 \cdot 10^{-92}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x 3.05e-109) (and (not (<= x 1.02e-102)) (<= x 3.2e-92)))
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= 3.05e-109) || (!(x <= 1.02e-102) && (x <= 3.2e-92))) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = 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) :: tmp
    if ((x <= 3.05d-109) .or. (.not. (x <= 1.02d-102)) .and. (x <= 3.2d-92)) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / ((x * 2.0d0) + (eps * ((-0.5d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= 3.05e-109) || (!(x <= 1.02e-102) && (x <= 3.2e-92))) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= 3.05e-109) or (not (x <= 1.02e-102) and (x <= 3.2e-92)):
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= 3.05e-109) || (!(x <= 1.02e-102) && (x <= 3.2e-92)))
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(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 <= 3.05e-109) || (~((x <= 1.02e-102)) && (x <= 3.2e-92)))
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, 3.05e-109], And[N[Not[LessEqual[x, 1.02e-102]], $MachinePrecision], LessEqual[x, 3.2e-92]]], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0499999999999998e-109 or 1.01999999999999996e-102 < x < 3.1999999999999997e-92

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0 93.4%

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

      1. neg-mul-1 93.4%

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

   5. Simplified 93.4%

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

   if 3.0499999999999998e-109 < x < 1.01999999999999996e-102 or 3.1999999999999997e-92 < x

   1. Initial program 21.9%

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

      1. flip-- 21.9%

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

      2. div-inv 21.9%

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

      3. add-sqr-sqrt 21.9%

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

      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 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 59.1%

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

      9. hypot-def 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. +-inverses 59.1%

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

      2. +-lft-identity 59.1%

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

      3. associate-*r/ 59.2%

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

      4. associate-/l* 59.2%

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

      5. /-rgt-identity 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 86.1%

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

      8. associate-*r* 86.1%

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

      9. metadata-eval 86.1%

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

      10. associate-*r/ 86.1%

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

      11. *-commutative 86.1%

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

   9. Simplified 86.1%

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

       1. fma-udef 86.1%

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

       2. associate-*l/ 86.1%

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

       3. associate-/l* 86.1%

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

       4. div-inv 86.1%

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

       5. clear-num 86.1%

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

   11. Applied egg-rr 86.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.05 \cdot 10^{-109} \lor \neg \left(x \leq 1.02 \cdot 10^{-102}\right) \land x \leq 3.2 \cdot 10^{-92}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

   1. flip-- 60.8%

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

   2. div-inv 60.7%

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

   3. add-sqr-sqrt 60.5%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 78.8%

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

   9. hypot-def 78.8%

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

4. Applied egg-rr 78.8%

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

   1. *-commutative 78.8%

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

   2. associate-/r/ 78.5%

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

   3. +-inverses 78.5%

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

   4. +-commutative 78.5%

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

6. Simplified 78.5%

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

7. Taylor expanded in x around inf 0.0%

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

   1. fma-def 0.0%

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

   2. metadata-eval 0.0%

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

   3. pow-sqr 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 0.0%

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

   8. metadata-eval 0.0%

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

   9. *-rgt-identity 0.0%

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

   10. +-commutative 0.0%

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

   11. fma-def 0.0%

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

   12. associate-*r/ 0.0%

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

   13. unpow2 0.0%

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

   14. rem-square-sqrt 43.3%

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

   15. metadata-eval 43.3%

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

9. Simplified 43.3%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{3}}, \mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)\right)}} \]
10. Step-by-step derivation

    1. fma-udef 43.3%

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

    2. fma-udef 43.3%

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

    3. associate-+r+ 43.3%

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

    4. *-commutative 43.3%

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

    5. div-inv 41.8%

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

    6. pow-flip 41.8%

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

    7. metadata-eval 41.8%

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

11. Applied egg-rr 41.8%

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

12. Taylor expanded in eps around 0 46.2%

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

    1. associate-*r/ 46.2%

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

    2. metadata-eval 46.2%

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

14. Simplified 46.2%

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

15. Final simplification 46.2%

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

Alternative 8: 45.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

   1. flip-- 60.8%

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

   2. div-inv 60.7%

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

   3. add-sqr-sqrt 60.5%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 78.8%

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

   9. hypot-def 78.8%

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

4. Applied egg-rr 78.8%

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

   1. +-inverses 78.8%

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

   2. +-lft-identity 78.8%

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

   3. associate-*r/ 78.8%

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

   4. associate-/l* 78.8%

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

   5. /-rgt-identity 78.8%

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

6. Simplified 78.8%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. *-commutative 0.0%

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

   3. fma-def 0.0%

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

   4. associate-*r/ 0.0%

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

   5. *-commutative 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 46.6%

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

   8. associate-*r* 46.6%

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

   9. metadata-eval 46.6%

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

   10. associate-*r/ 46.6%

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

   11. *-commutative 46.6%

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

9. Simplified 46.6%

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

    1. fma-udef 46.6%

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

    2. associate-*l/ 46.6%

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

    3. associate-/l* 46.6%

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

    4. div-inv 46.6%

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

    5. clear-num 46.6%

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

11. Applied egg-rr 46.6%

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

12. Final simplification 46.6%

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

Alternative 9: 44.9% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

   1. flip-- 60.8%

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

   2. div-inv 60.7%

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

   3. add-sqr-sqrt 60.5%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 78.8%

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

   9. hypot-def 78.8%

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

4. Applied egg-rr 78.8%

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

   1. +-inverses 78.8%

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

   2. +-lft-identity 78.8%

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

   3. associate-*r/ 78.8%

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

   4. associate-/l* 78.8%

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

   5. /-rgt-identity 78.8%

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

6. Simplified 78.8%

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

7. Taylor expanded in eps around 0 45.6%

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

   1. associate-*r/ 45.6%

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

   2. associate-/l* 45.2%

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

9. Simplified 45.2%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{x}{\varepsilon}}} \]
10. Step-by-step derivation

    1. associate-/r/ 45.5%

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

11. Applied egg-rr 45.5%

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

12. Final simplification 45.5%

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

Alternative 10: 45.1% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in x around inf 45.6%

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

   1. *-commutative 45.6%

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

   2. associate-*l/ 45.6%

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

5. Simplified 45.6%

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

6. Final simplification 45.6%

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

Alternative 11: 5.3% 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 60.9%

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

   1. flip-- 60.8%

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

   2. div-inv 60.7%

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

   3. add-sqr-sqrt 60.5%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 78.8%

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

   9. hypot-def 78.8%

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

4. Applied egg-rr 78.8%

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

   1. +-inverses 78.8%

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

   2. +-lft-identity 78.8%

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

   3. associate-*r/ 78.8%

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

   4. associate-/l* 78.8%

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

   5. /-rgt-identity 78.8%

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

6. Simplified 78.8%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. *-commutative 0.0%

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

   3. fma-def 0.0%

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

   4. associate-*r/ 0.0%

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

   5. *-commutative 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 46.6%

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

   8. associate-*r* 46.6%

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

   9. metadata-eval 46.6%

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

   10. associate-*r/ 46.6%

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

   11. *-commutative 46.6%

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

9. Simplified 46.6%

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

10. Taylor expanded in eps around inf 5.4%

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

    1. *-commutative 5.4%

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

12. Simplified 5.4%

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

13. Final simplification 5.4%

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

Alternative 12: 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 60.9%

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

   1. sub-neg 60.9%

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

   2. +-commutative 60.9%

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

   3. add-sqr-sqrt 60.3%

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

   4. distribute-rgt-neg-in 60.3%

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

   5. fma-def 60.0%

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

   6. pow1/2 60.0%

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

   7. sqrt-pow1 60.2%

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

   8. pow2 60.2%

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

   9. metadata-eval 60.2%

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

   10. pow1/2 60.2%

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

   11. sqrt-pow1 60.0%

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

   12. pow2 60.0%

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

   13. metadata-eval 60.0%

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

4. Applied egg-rr 60.0%

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

5. Taylor expanded in x around inf 4.4%

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

   1. distribute-rgt1-in 4.4%

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

   2. metadata-eval 4.4%

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

   3. mul0-lft 4.4%

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

7. Simplified 4.4%

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

8. Final simplification 4.4%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(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))))