ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.3% → 99.0%

Time: 9.7s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% 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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, {\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154)
     t_0
     (/
      eps
      (+
       x
       (+ x (fma -0.125 (pow (/ eps (pow x 1.5)) 2.0) (* (/ eps x) -0.5))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + fma(-0.125, pow((eps / pow(x, 1.5)), 2.0), ((eps / x) * -0.5))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + fma(-0.125, (Float64(eps / (x ^ 1.5)) ^ 2.0), 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, -1e-154], t$95$0, N[(eps / N[(x + N[(x + N[(-0.125 * N[Power[N[(eps / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $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))) < -9.9999999999999997e-155

   1. Initial program 99.3%

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

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

   1. Initial program 8.4%

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

      1. flip-- 8.4%

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

      2. div-inv 8.4%

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

      3. add-sqr-sqrt 8.5%

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

      4. associate--r- 99.6%

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

      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 45.8%

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

      9. hypot-def 45.8%

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

   4. Applied egg-rr 45.8%

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

      1. +-inverses 45.8%

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

      2. +-lft-identity 45.8%

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

      3. associate-*r/ 45.9%

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

      4. associate-/l* 45.9%

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

      5. /-rgt-identity 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \left(-0.125 \cdot \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{4}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)}} \]
   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. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. associate-*r/ 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 96.8%

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

      15. associate-*r* 96.8%

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

      16. metadata-eval 96.8%

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

      17. associate-*r/ 96.8%

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

      18. *-commutative 96.8%

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

   9. Simplified 96.8%

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

       1. add-sqr-sqrt 96.8%

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

       2. sqrt-div 96.8%

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

       3. unpow2 96.8%

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

       4. sqrt-prod 52.9%

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

       5. add-sqr-sqrt 96.8%

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

       6. sqrt-pow1 96.8%

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

       7. metadata-eval 96.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{\color{blue}{1.5}}} \cdot \sqrt{\frac{{\varepsilon}^{2}}{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       8. sqrt-div 96.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \color{blue}{\frac{\sqrt{{\varepsilon}^{2}}}{\sqrt{{x}^{3}}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       9. unpow2 96.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}}{\sqrt{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       10. sqrt-prod 53.1%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}}{\sqrt{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       11. add-sqr-sqrt 97.0%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\color{blue}{\varepsilon}}{\sqrt{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       12. sqrt-pow1 100.0%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\varepsilon}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       13. metadata-eval 100.0%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\varepsilon}{{x}^{\color{blue}{1.5}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{\frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\varepsilon}{{x}^{1.5}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]
   12. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{{\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{{\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, {\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + \left(-0.125 \cdot {\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2} + x \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154)
     t_0
     (/
      eps
      (+
       (* (/ eps x) -0.5)
       (+ (* -0.125 (pow (/ eps (pow x 1.5)) 2.0)) (* x 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (((eps / x) * -0.5) + ((-0.125 * pow((eps / pow(x, 1.5)), 2.0)) + (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-154)) then
        tmp = t_0
    else
        tmp = eps / (((eps / x) * (-0.5d0)) + (((-0.125d0) * ((eps / (x ** 1.5d0)) ** 2.0d0)) + (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (((eps / x) * -0.5) + ((-0.125 * Math.pow((eps / Math.pow(x, 1.5)), 2.0)) + (x * 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-154:
		tmp = t_0
	else:
		tmp = eps / (((eps / x) * -0.5) + ((-0.125 * math.pow((eps / math.pow(x, 1.5)), 2.0)) + (x * 2.0)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(Float64(eps / x) * -0.5) + Float64(Float64(-0.125 * (Float64(eps / (x ^ 1.5)) ^ 2.0)) + Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = eps / (((eps / x) * -0.5) + ((-0.125 * ((eps / (x ^ 1.5)) ^ 2.0)) + (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-154], t$95$0, N[(eps / N[(N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[(-0.125 * N[Power[N[(eps / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.3%

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

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

   1. Initial program 8.4%

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

      1. flip-- 8.4%

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

      2. div-inv 8.4%

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

      3. add-sqr-sqrt 8.5%

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

      4. associate--r- 99.6%

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

      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 45.8%

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

      9. hypot-def 45.8%

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

   4. Applied egg-rr 45.8%

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

      1. +-inverses 45.8%

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

      2. +-lft-identity 45.8%

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

      3. associate-*r/ 45.9%

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

      4. associate-/l* 45.9%

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

      5. /-rgt-identity 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \left(-0.125 \cdot \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{4}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)}} \]
   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. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. associate-*r/ 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 96.8%

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

      15. associate-*r* 96.8%

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

      16. metadata-eval 96.8%

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

      17. associate-*r/ 96.8%

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

      18. *-commutative 96.8%

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

   9. Simplified 96.8%

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

   10. Taylor expanded in x around 0 96.8%

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

       1. add-sqr-sqrt 96.8%

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

       2. sqrt-div 96.8%

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

       3. unpow2 96.8%

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

       4. sqrt-prod 52.9%

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

       5. add-sqr-sqrt 96.8%

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

       6. sqrt-pow1 96.8%

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

       7. metadata-eval 96.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{\color{blue}{1.5}}} \cdot \sqrt{\frac{{\varepsilon}^{2}}{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       8. sqrt-div 96.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \color{blue}{\frac{\sqrt{{\varepsilon}^{2}}}{\sqrt{{x}^{3}}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       9. unpow2 96.8%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}}{\sqrt{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       10. sqrt-prod 53.1%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}}{\sqrt{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       11. add-sqr-sqrt 97.0%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\color{blue}{\varepsilon}}{\sqrt{{x}^{3}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       12. sqrt-pow1 100.0%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\varepsilon}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

       13. metadata-eval 100.0%

           \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\varepsilon}{{x}^{\color{blue}{1.5}}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + \left(-0.125 \cdot \color{blue}{\left(\frac{\varepsilon}{{x}^{1.5}} \cdot \frac{\varepsilon}{{x}^{1.5}}\right)} + 2 \cdot x\right)} \]
   13. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \color{blue}{{\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2}}, \frac{\varepsilon}{x} \cdot -0.5\right)\right)} \]

   14. Simplified 99.9%

       \[\leadsto \frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + \left(-0.125 \cdot \color{blue}{{\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2}} + 2 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + \left(-0.125 \cdot {\left(\frac{\varepsilon}{{x}^{1.5}}\right)}^{2} + x \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.3%

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

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

   1. Initial program 8.4%

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

      1. flip-- 8.4%

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

      2. div-inv 8.4%

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

      3. add-sqr-sqrt 8.5%

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

      4. associate--r- 99.6%

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

      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 45.8%

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

      9. hypot-def 45.8%

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

   4. Applied egg-rr 45.8%

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

      1. +-inverses 45.8%

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

      2. +-lft-identity 45.8%

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

      3. associate-*r/ 45.9%

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

      4. associate-/l* 45.9%

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

      5. /-rgt-identity 45.9%

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

   6. Simplified 45.9%

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

      1. hypot-udef 45.9%

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

      2. unpow2 45.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. sub-neg 100.0%

         \[\leadsto \frac{\varepsilon}{x + \sqrt{\color{blue}{{x}^{2} - \varepsilon}}} \]

      5. flip3-- 67.4%

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

      6. sqrt-div 67.5%

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

   8. Applied egg-rr 67.4%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\frac{\sqrt{{x}^{6} + {\varepsilon}^{3}}}{\sqrt{{x}^{4} + \varepsilon \cdot \mathsf{fma}\left(x, x, \varepsilon\right)}}}} \]

   9. Taylor expanded in x around inf 99.6%

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

4. Final simplification 99.4%

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

Alternative 4: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-96}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.05e-96)
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (* (/ eps x) -0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.05e-96) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.05d-96) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + ((eps / x) * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.05e-96) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.05e-96:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + ((eps / x) * -0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.05e-96)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(eps / x) * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.05e-96)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.05e-96], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.05000000000000012e-96

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0 96.9%

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

      1. neg-mul-1 96.9%

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

   5. Simplified 96.9%

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

   if 2.05000000000000012e-96 < x

   1. Initial program 26.7%

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

      1. flip-- 26.7%

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

      2. div-inv 26.6%

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

      3. add-sqr-sqrt 26.6%

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

      4. associate--r- 99.5%

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

      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 56.4%

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

      9. hypot-def 56.4%

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

   4. Applied egg-rr 56.4%

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

      1. +-inverses 56.4%

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

      2. +-lft-identity 56.4%

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

      3. associate-*r/ 56.5%

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

      4. associate-/l* 56.5%

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

      5. /-rgt-identity 56.5%

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

   6. Simplified 56.5%

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

      1. hypot-udef 56.5%

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

      2. unpow2 56.5%

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

      3. add-sqr-sqrt 99.9%

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

      4. sub-neg 99.9%

         \[\leadsto \frac{\varepsilon}{x + \sqrt{\color{blue}{{x}^{2} - \varepsilon}}} \]

      5. flip3-- 72.0%

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

      6. sqrt-div 72.0%

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

   8. Applied egg-rr 56.2%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\frac{\sqrt{{x}^{6} + {\varepsilon}^{3}}}{\sqrt{{x}^{4} + \varepsilon \cdot \mathsf{fma}\left(x, x, \varepsilon\right)}}}} \]

   9. Taylor expanded in x around inf 82.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-96}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. flip-- 64.5%

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

   2. div-inv 64.3%

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

   3. add-sqr-sqrt 64.1%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 78.8%

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

   9. hypot-def 78.8%

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

4. Applied egg-rr 78.8%

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

   1. +-inverses 78.8%

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

   2. +-lft-identity 78.8%

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

   3. associate-*r/ 78.9%

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

   4. associate-/l* 78.9%

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

   5. /-rgt-identity 78.9%

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

6. Simplified 78.9%

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

   1. hypot-udef 78.9%

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

   2. unpow2 78.9%

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

   3. add-sqr-sqrt 99.6%

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

   4. sub-neg 99.6%

      \[\leadsto \frac{\varepsilon}{x + \sqrt{\color{blue}{{x}^{2} - \varepsilon}}} \]

   5. flip3-- 52.1%

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

   6. sqrt-div 52.1%

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

8. Applied egg-rr 26.9%

   \[\leadsto \frac{\varepsilon}{x + \color{blue}{\frac{\sqrt{{x}^{6} + {\varepsilon}^{3}}}{\sqrt{{x}^{4} + \varepsilon \cdot \mathsf{fma}\left(x, x, \varepsilon\right)}}}} \]

9. Taylor expanded in x around inf 42.6%

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

10. Final simplification 42.6%

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

Alternative 6: 44.0% 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 64.5%

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

3. Taylor expanded in x around inf 41.8%

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

4. Final simplification 41.8%

   \[\leadsto \frac{\varepsilon}{x} \cdot 0.5 \]
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 64.5%

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

   1. flip-- 64.5%

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

   2. div-inv 64.3%

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

   3. add-sqr-sqrt 64.1%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 78.8%

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

   9. hypot-def 78.8%

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

4. Applied egg-rr 78.8%

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

   1. +-inverses 78.8%

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

   2. +-lft-identity 78.8%

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

   3. associate-*r/ 78.9%

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

   4. associate-/l* 78.9%

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

   5. /-rgt-identity 78.9%

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

6. Simplified 78.9%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 42.6%

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

   6. mul-1-neg 42.6%

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

9. Simplified 42.6%

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

10. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

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

12. Simplified 5.3%

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

13. Final simplification 5.3%

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

Alternative 8: 4.3% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 64.5%

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

   1. sub-neg 64.5%

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

   2. +-commutative 64.5%

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

   3. add-sqr-sqrt 63.9%

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

   4. distribute-rgt-neg-in 63.9%

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

   5. fma-def 63.6%

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

   6. pow1/2 63.6%

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

   7. sqrt-pow1 63.8%

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

   8. pow2 63.8%

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

   9. metadata-eval 63.8%

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

   10. pow1/2 63.8%

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

   11. sqrt-pow1 63.7%

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

   12. pow2 63.7%

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

   13. metadata-eval 63.7%

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

4. Applied egg-rr 63.7%

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

5. Taylor expanded in x around inf 4.3%

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

   1. distribute-rgt1-in 4.3%

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

   2. metadata-eval 4.3%

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

   3. mul0-lft 4.3%

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

7. Simplified 4.3%

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

8. Final simplification 4.3%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(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))))