ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.1% → 99.3%

Time: 10.4s

Alternatives: 7

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: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

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(-0.125, (((eps / x) / (x / eps)) / x), ((eps / x) * -0.5))));
	}
	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(-0.125, Float64(Float64(Float64(eps / x) / Float64(x / eps)) / x), Float64(Float64(eps / x) * -0.5)))));
	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[(-0.125 * N[(N[(N[(eps / x), $MachinePrecision] / N[(x / eps), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $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 98.6%

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

      1. flip-- 98.5%

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

      2. div-inv 98.2%

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

      3. add-sqr-sqrt 98.1%

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

      4. associate--r- 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 99.0%

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

      6. pow2 99.3%

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

      7. sub-neg 99.3%

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

      8. add-sqr-sqrt 99.3%

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

      9. hypot-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/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. /-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

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

   1. Initial program 9.3%

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

      1. flip-- 9.3%

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

      2. div-inv 9.3%

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

      3. add-sqr-sqrt 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 44.9%

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

      9. hypot-def 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. +-inverses 44.9%

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

      2. +-lft-identity 44.9%

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

      3. associate-*r/ 45.0%

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

      4. associate-/l* 45.0%

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

      5. /-rgt-identity 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. metadata-eval 0.0%

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

      3. pow-sqr 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 0.0%

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

      8. metadata-eval 0.0%

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

      9. *-rgt-identity 0.0%

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

      10. associate-*r/ 0.0%

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

      11. *-commutative 0.0%

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

      12. unpow2 0.0%

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

      13. rem-square-sqrt 89.4%

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

      14. associate-*r* 89.4%

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

      15. metadata-eval 89.4%

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

      16. associate-*r/ 89.4%

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

      17. *-commutative 89.4%

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

   9. Simplified 89.4%

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

       1. *-un-lft-identity 89.4%

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

       2. cube-mult 89.4%

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

       3. times-frac 99.2%

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

       4. unpow2 99.2%

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

       5. frac-times 99.3%

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

       6. pow2 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. associate-*l/ 99.3%

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

       2. *-lft-identity 99.3%

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

   13. Simplified 99.3%

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

       1. unpow2 99.3%

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

       2. clear-num 99.3%

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

       3. un-div-inv 99.3%

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

   15. Applied egg-rr 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

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 -0.125 (/ (/ (/ eps x) (/ x eps)) x) (* (/ eps x) -0.5))))))))

C

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

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -2e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + fma(-0.125, Float64(Float64(Float64(eps / x) / Float64(x / eps)) / x), Float64(Float64(eps / x) * -0.5)))));
	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[(-0.125 * N[(N[(N[(eps / x), $MachinePrecision] / N[(x / eps), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $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 98.6%

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

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

   1. Initial program 9.3%

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

      1. flip-- 9.3%

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

      2. div-inv 9.3%

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

      3. add-sqr-sqrt 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 44.9%

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

      9. hypot-def 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. +-inverses 44.9%

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

      2. +-lft-identity 44.9%

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

      3. associate-*r/ 45.0%

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

      4. associate-/l* 45.0%

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

      5. /-rgt-identity 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. metadata-eval 0.0%

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

      3. pow-sqr 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 0.0%

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

      8. metadata-eval 0.0%

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

      9. *-rgt-identity 0.0%

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

      10. associate-*r/ 0.0%

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

      11. *-commutative 0.0%

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

      12. unpow2 0.0%

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

      13. rem-square-sqrt 89.4%

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

      14. associate-*r* 89.4%

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

      15. metadata-eval 89.4%

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

      16. associate-*r/ 89.4%

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

      17. *-commutative 89.4%

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

   9. Simplified 89.4%

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

       1. *-un-lft-identity 89.4%

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

       2. cube-mult 89.4%

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

       3. times-frac 99.2%

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

       4. unpow2 99.2%

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

       5. frac-times 99.3%

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

       6. pow2 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. associate-*l/ 99.3%

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

       2. *-lft-identity 99.3%

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

   13. Simplified 99.3%

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

       1. unpow2 99.3%

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

       2. clear-num 99.3%

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

       3. un-div-inv 99.3%

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

   15. Applied egg-rr 99.3%

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

4. Final simplification 98.9%

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

Alternative 3: 99.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154) 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-154) {
		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-154)) 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-154) {
		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-154:
		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-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(Float64(eps * -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 * 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-154], t$95$0, N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.6%

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

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

   1. Initial program 9.3%

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

      1. flip-- 9.3%

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

      2. div-inv 9.3%

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

      3. add-sqr-sqrt 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 44.9%

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

      9. hypot-def 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. +-inverses 44.9%

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

      2. +-lft-identity 44.9%

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

      3. associate-*r/ 45.0%

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

      4. associate-/l* 45.0%

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

      5. /-rgt-identity 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 99.0%

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

      8. associate-*r* 99.0%

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

      9. metadata-eval 99.0%

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

      10. associate-*r/ 99.0%

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

      11. *-commutative 99.0%

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

   9. Simplified 99.0%

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

       1. fma-udef 99.0%

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

       2. associate-*l/ 99.0%

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

   11. Applied egg-rr 99.0%

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

4. Final simplification 98.8%

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

Alternative 4: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-114} \lor \neg \left(x \leq 5 \cdot 10^{-91}\right) \land x \leq 9.2 \cdot 10^{-66}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x 1.9e-114) (and (not (<= x 5e-91)) (<= x 9.2e-66)))
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (/ (* eps -0.5) x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= 1.9e-114) || (!(x <= 5e-91) && (x <= 9.2e-66))) {
		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 <= 1.9d-114) .or. (.not. (x <= 5d-91)) .and. (x <= 9.2d-66)) 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 <= 1.9e-114) || (!(x <= 5e-91) && (x <= 9.2e-66))) {
		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 <= 1.9e-114) or (not (x <= 5e-91) and (x <= 9.2e-66)):
		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 <= 1.9e-114) || (!(x <= 5e-91) && (x <= 9.2e-66)))
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(Float64(eps * -0.5) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 1.9e-114) || (~((x <= 5e-91)) && (x <= 9.2e-66)))
		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, 1.9e-114], And[N[Not[LessEqual[x, 5e-91]], $MachinePrecision], LessEqual[x, 9.2e-66]]], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999e-114 or 4.99999999999999997e-91 < x < 9.19999999999999967e-66

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0 86.8%

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

      1. neg-mul-1 86.8%

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

   5. Simplified 86.8%

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

   if 1.8999999999999999e-114 < x < 4.99999999999999997e-91 or 9.19999999999999967e-66 < x

   1. Initial program 21.3%

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

      1. flip-- 21.4%

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

      2. div-inv 21.3%

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

      3. add-sqr-sqrt 21.6%

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

      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 52.3%

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

      9. hypot-def 52.3%

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

   4. Applied egg-rr 52.3%

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

      1. +-inverses 52.3%

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

      2. +-lft-identity 52.3%

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

      3. associate-*r/ 52.5%

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

      4. associate-/l* 52.5%

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

      5. /-rgt-identity 52.5%

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

   6. Simplified 52.5%

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

   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 87.2%

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

      8. associate-*r* 87.2%

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

      9. metadata-eval 87.2%

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

      10. associate-*r/ 87.2%

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

      11. *-commutative 87.2%

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

   9. Simplified 87.2%

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

       1. fma-udef 87.2%

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

       2. associate-*l/ 87.2%

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

   11. Applied egg-rr 87.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-114} \lor \neg \left(x \leq 5 \cdot 10^{-91}\right) \land x \leq 9.2 \cdot 10^{-66}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.9% accurate, 9.7× speedup

Math

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

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(Float64(eps * -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[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

   1. flip-- 59.8%

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

   2. div-inv 59.7%

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

   3. add-sqr-sqrt 59.7%

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

   4. associate--r- 99.4%

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

   5. pow2 99.3%

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

   6. pow2 99.4%

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

   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 75.7%

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

   9. hypot-def 75.7%

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

4. Applied egg-rr 75.7%

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

   1. +-inverses 75.7%

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

   2. +-lft-identity 75.7%

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

   3. associate-*r/ 75.8%

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

   4. associate-/l* 75.8%

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

   5. /-rgt-identity 75.8%

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

6. Simplified 75.8%

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

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 48.0%

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

   8. associate-*r* 48.0%

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

   9. metadata-eval 48.0%

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

   10. associate-*r/ 48.0%

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

   11. *-commutative 48.0%

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

9. Simplified 48.0%

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

    1. fma-udef 48.0%

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

    2. associate-*l/ 48.0%

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

11. Applied egg-rr 48.0%

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

12. Final simplification 48.0%

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

Alternative 6: 44.2% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in x around inf 47.0%

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

4. Final simplification 47.0%

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

Alternative 7: 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 59.9%

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

   1. flip-- 59.8%

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

   2. div-inv 59.7%

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

   3. add-sqr-sqrt 59.7%

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

   4. associate--r- 99.4%

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

   5. pow2 99.3%

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

   6. pow2 99.4%

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

   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 75.7%

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

   9. hypot-def 75.7%

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

4. Applied egg-rr 75.7%

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

   1. +-inverses 75.7%

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

   2. +-lft-identity 75.7%

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

   3. associate-*r/ 75.8%

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

   4. associate-/l* 75.8%

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

   5. /-rgt-identity 75.8%

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

6. Simplified 75.8%

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

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 48.0%

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

   8. associate-*r* 48.0%

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

   9. metadata-eval 48.0%

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

   10. associate-*r/ 48.0%

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

   11. *-commutative 48.0%

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

9. Simplified 48.0%

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

10. Taylor expanded in eps around inf 5.6%

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

    1. *-commutative 5.6%

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

12. Simplified 5.6%

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

13. Final simplification 5.6%

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

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 2024031 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4d"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))

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

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