ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.3% → 99.2%

Time: 12.1s

Alternatives: 9

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: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/
    eps
    (+
     x
     (+ x (fma (/ eps x) -0.5 (/ -0.125 (* x (/ (/ x eps) (/ eps x))))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / (x + (x + fma((eps / x), -0.5, (-0.125 / (x * ((x / eps) / (eps / x)))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(x + Float64(x + fma(Float64(eps / x), -0.5, Float64(-0.125 / Float64(x * Float64(Float64(x / eps) / Float64(eps / 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-154], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5 + N[(-0.125 / N[(x * N[(N[(x / eps), $MachinePrecision] / N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

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

      1. flip-- 97.6%

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

      2. div-inv 97.3%

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

      3. add-sqr-sqrt 97.2%

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

      4. associate--r- 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. sub-neg 99.2%

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

      8. add-sqr-sqrt 99.2%

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

      9. hypot-def 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. +-inverses 99.2%

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

      2. +-lft-identity 99.2%

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

      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-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 10.0%

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

      1. flip-- 9.9%

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

      2. div-inv 9.9%

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

      3. add-sqr-sqrt 10.0%

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

      4. associate--r- 99.6%

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

      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 46.2%

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

      9. hypot-def 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. +-inverses 46.2%

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

      2. +-lft-identity 46.2%

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

      3. associate-*r/ 46.4%

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

      4. associate-/l* 46.4%

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

      5. /-rgt-identity 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in x around inf 0.0%

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

   9. Simplified 88.7%

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

       1. cube-mult 88.7%

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

       2. *-un-lft-identity 88.7%

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

       3. times-frac 99.0%

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

       4. unpow2 99.0%

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

       5. frac-times 99.2%

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

       6. pow2 99.2%

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

   11. Applied egg-rr 99.2%

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

       1. /-rgt-identity 99.2%

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

   13. Simplified 99.2%

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

       1. unpow2 99.2%

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

       2. clear-num 99.2%

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

       3. un-div-inv 99.2%

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

   15. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (hypot (sqrt (- eps)) x))
   (/
    eps
    (+
     x
     (+ x (fma (/ eps x) -0.5 (/ -0.125 (* x (/ (/ x eps) (/ eps x))))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / (x + (x + fma((eps / x), -0.5, (-0.125 / (x * ((x / eps) / (eps / x)))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / Float64(x + Float64(x + fma(Float64(eps / x), -0.5, Float64(-0.125 / Float64(x * Float64(Float64(x / eps) / Float64(eps / 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-154], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5 + N[(-0.125 / N[(x * N[(N[(x / eps), $MachinePrecision] / N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

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

      1. sub-neg 97.8%

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

      2. +-commutative 97.8%

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

      3. add-sqr-sqrt 97.7%

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

      4. hypot-def 97.8%

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

   4. Applied egg-rr 97.8%

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

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

   1. Initial program 10.0%

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

      1. flip-- 9.9%

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

      2. div-inv 9.9%

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

      3. add-sqr-sqrt 10.0%

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

      4. associate--r- 99.6%

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

      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 46.2%

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

      9. hypot-def 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. +-inverses 46.2%

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

      2. +-lft-identity 46.2%

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

      3. associate-*r/ 46.4%

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

      4. associate-/l* 46.4%

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

      5. /-rgt-identity 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in x around inf 0.0%

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

   9. Simplified 88.7%

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

       1. cube-mult 88.7%

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

       2. *-un-lft-identity 88.7%

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

       3. times-frac 99.0%

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

       4. unpow2 99.0%

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

       5. frac-times 99.2%

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

       6. pow2 99.2%

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

   11. Applied egg-rr 99.2%

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

       1. /-rgt-identity 99.2%

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

   13. Simplified 99.2%

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

       1. unpow2 99.2%

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

       2. clear-num 99.2%

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

       3. un-div-inv 99.2%

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

   15. Applied egg-rr 99.2%

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

4. Final simplification 98.4%

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

Alternative 3: 99.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154)
     t_0
     (/
      eps
      (+
       x
       (+ x (fma (/ eps x) -0.5 (/ -0.125 (* x (/ (/ x eps) (/ eps x)))))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + fma((eps / x), -0.5, (-0.125 / (x * ((x / eps) / (eps / 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-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + fma(Float64(eps / x), -0.5, Float64(-0.125 / Float64(x * Float64(Float64(x / eps) / Float64(eps / x))))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-154], t$95$0, N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5 + N[(-0.125 / N[(x * N[(N[(x / eps), $MachinePrecision] / N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

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

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

   1. Initial program 10.0%

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

      1. flip-- 9.9%

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

      2. div-inv 9.9%

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

      3. add-sqr-sqrt 10.0%

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

      4. associate--r- 99.6%

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

      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 46.2%

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

      9. hypot-def 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. +-inverses 46.2%

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

      2. +-lft-identity 46.2%

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

      3. associate-*r/ 46.4%

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

      4. associate-/l* 46.4%

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

      5. /-rgt-identity 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in x around inf 0.0%

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

   9. Simplified 88.7%

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

       1. cube-mult 88.7%

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

       2. *-un-lft-identity 88.7%

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

       3. times-frac 99.0%

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

       4. unpow2 99.0%

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

       5. frac-times 99.2%

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

       6. pow2 99.2%

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

   11. Applied egg-rr 99.2%

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

       1. /-rgt-identity 99.2%

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

   13. Simplified 99.2%

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

       1. unpow2 99.2%

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

       2. clear-num 99.2%

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

       3. un-div-inv 99.2%

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

   15. Applied egg-rr 99.2%

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

4. Final simplification 98.4%

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

Alternative 4: 98.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154) t_0 (/ eps (+ x (+ x (* 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-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (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-154)) then
        tmp = t_0
    else
        tmp = eps / (x + (x + (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-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (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-154:
		tmp = t_0
	else:
		tmp = eps / (x + (x + (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-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + 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-154)
		tmp = t_0;
	else
		tmp = eps / (x + (x + (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-154], t$95$0, N[(eps / N[(x + N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

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

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

   1. Initial program 10.0%

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

      1. flip-- 9.9%

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

      2. div-inv 9.9%

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

      3. add-sqr-sqrt 10.0%

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

      4. associate--r- 99.6%

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

      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 46.2%

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

      9. hypot-def 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. +-inverses 46.2%

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

      2. +-lft-identity 46.2%

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

      3. associate-*r/ 46.4%

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

      4. associate-/l* 46.4%

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

      5. /-rgt-identity 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l/ 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 98.9%

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

      7. associate-*r* 98.9%

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

      8. metadata-eval 98.9%

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

      9. associate-/l* 98.9%

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

   9. Simplified 98.9%

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

       1. associate-/r/ 98.9%

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

   11. Applied egg-rr 98.9%

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

4. Final simplification 98.2%

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

Alternative 5: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-130} \lor \neg \left(x \leq 2.9 \cdot 10^{-106}\right) \land x \leq 6 \cdot 10^{-82}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x 6.5e-130) (and (not (<= x 2.9e-106)) (<= x 6e-82)))
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (* eps (/ -0.5 x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= 6.5e-130) || (!(x <= 2.9e-106) && (x <= 6e-82))) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (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 <= 6.5d-130) .or. (.not. (x <= 2.9d-106)) .and. (x <= 6d-82)) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + (eps * ((-0.5d0) / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= 6.5e-130) || (!(x <= 2.9e-106) && (x <= 6e-82))) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (eps * (-0.5 / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= 6.5e-130) or (not (x <= 2.9e-106) and (x <= 6e-82)):
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + (eps * (-0.5 / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= 6.5e-130) || (!(x <= 2.9e-106) && (x <= 6e-82)))
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(eps * Float64(-0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 6.5e-130) || (~((x <= 2.9e-106)) && (x <= 6e-82)))
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + (eps * (-0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, 6.5e-130], And[N[Not[LessEqual[x, 2.9e-106]], $MachinePrecision], LessEqual[x, 6e-82]]], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5000000000000002e-130 or 2.9e-106 < x < 5.9999999999999998e-82

   1. Initial program 96.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 6.5000000000000002e-130 < x < 2.9e-106 or 5.9999999999999998e-82 < x

   1. Initial program 18.9%

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

      1. flip-- 18.8%

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

      2. div-inv 18.8%

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

      3. add-sqr-sqrt 19.0%

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

      4. associate--r- 99.5%

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

      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 55.8%

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

      9. hypot-def 55.8%

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

   4. Applied egg-rr 55.8%

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

      1. +-inverses 55.8%

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

      2. +-lft-identity 55.8%

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

      3. associate-*r/ 56.0%

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

      4. associate-/l* 56.0%

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

      5. /-rgt-identity 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l/ 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 91.3%

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

      7. associate-*r* 91.3%

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

      8. metadata-eval 91.3%

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

      9. associate-/l* 91.3%

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

   9. Simplified 91.3%

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

       1. associate-/r/ 91.3%

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

   11. Applied egg-rr 91.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-130} \lor \neg \left(x \leq 2.9 \cdot 10^{-106}\right) \land x \leq 6 \cdot 10^{-82}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \varepsilon \cdot \frac{-0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. flip-- 59.3%

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

   2. div-inv 59.1%

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

   3. add-sqr-sqrt 59.0%

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

   4. associate--r- 99.4%

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

   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 76.0%

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

   9. hypot-def 76.0%

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

4. Applied egg-rr 76.0%

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

   1. +-inverses 76.0%

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

   2. +-lft-identity 76.0%

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

   3. associate-*r/ 76.1%

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

   4. associate-/l* 76.1%

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

   5. /-rgt-identity 76.1%

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

6. Simplified 76.1%

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

7. Taylor expanded in x around inf 0.0%

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

   1. *-commutative 0.0%

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

   2. associate-*l/ 0.0%

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

   3. *-commutative 0.0%

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

   4. *-commutative 0.0%

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

   5. unpow2 0.0%

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

   6. rem-square-sqrt 48.9%

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

   7. associate-*r* 48.9%

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

   8. metadata-eval 48.9%

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

   9. associate-/l* 48.9%

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

9. Simplified 48.9%

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

    1. associate-/r/ 48.9%

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

11. Applied egg-rr 48.9%

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

12. Final simplification 48.9%

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

Alternative 7: 44.8% 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 59.4%

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

   1. sub-neg 59.4%

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

   2. +-commutative 59.4%

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

   3. add-sqr-sqrt 56.1%

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

   4. hypot-def 56.1%

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

4. Applied egg-rr 56.1%

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

5. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-rgt-identity 0.0%

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

   3. times-frac 0.0%

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

   4. *-commutative 0.0%

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

   5. unpow2 0.0%

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

   6. rem-square-sqrt 47.3%

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

   7. neg-mul-1 47.3%

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

   8. distribute-neg-frac 47.3%

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

   9. /-rgt-identity 47.3%

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

   10. distribute-rgt-neg-in 47.3%

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

   11. distribute-lft-neg-out 47.3%

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

   12. *-commutative 47.3%

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

   13. distribute-neg-frac 47.3%

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

   14. metadata-eval 47.3%

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

7. Simplified 47.3%

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

8. Final simplification 47.3%

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

Alternative 8: 45.0% 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 59.4%

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

3. Taylor expanded in x around inf 47.4%

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

   1. *-commutative 47.4%

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

   2. associate-*l/ 47.4%

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

5. Simplified 47.4%

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

6. Final simplification 47.4%

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

Alternative 9: 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 59.4%

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

   1. flip-- 59.3%

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

   2. div-inv 59.1%

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

   3. add-sqr-sqrt 59.0%

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

   4. associate--r- 99.4%

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

   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 76.0%

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

   9. hypot-def 76.0%

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

4. Applied egg-rr 76.0%

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

   1. +-inverses 76.0%

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

   2. +-lft-identity 76.0%

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

   3. associate-*r/ 76.1%

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

   4. associate-/l* 76.1%

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

   5. /-rgt-identity 76.1%

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

6. Simplified 76.1%

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

7. Taylor expanded in x around inf 0.0%

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

   1. *-commutative 0.0%

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

   2. associate-*l/ 0.0%

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

   3. *-commutative 0.0%

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

   4. *-commutative 0.0%

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

   5. unpow2 0.0%

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

   6. rem-square-sqrt 48.9%

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

   7. associate-*r* 48.9%

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

   8. metadata-eval 48.9%

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

   9. associate-/l* 48.9%

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

9. Simplified 48.9%

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

10. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

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

12. Simplified 5.3%

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

13. Final simplification 5.3%

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

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024017 
(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))))