ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.1% → 98.9%

Time: 6.9s

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: 62.1% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-152)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/
    eps
    (fma
     -0.125
     (* (/ eps (* x x)) (/ eps x))
     (+ (* (/ eps x) -0.5) (* x 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-152) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / fma(-0.125, ((eps / (x * x)) * (eps / x)), (((eps / x) * -0.5) + (x * 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-152)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / fma(-0.125, Float64(Float64(eps / Float64(x * x)) * Float64(eps / x)), Float64(Float64(Float64(eps / x) * -0.5) + Float64(x * 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-152], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(-0.125 * N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps / x), $MachinePrecision] * -0.5), $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))) < -1.00000000000000007e-152

   1. Initial program 98.8%

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

      1. flip-- 98.7%

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

      2. div-inv 98.5%

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

      3. add-sqr-sqrt 98.0%

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

      4. sub-neg 98.0%

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

      5. add-sqr-sqrt 98.0%

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

      6. hypot-def 98.0%

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

   3. Applied egg-rr 98.0%

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

      1. associate-*r/ 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate--r- 99.1%

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

      4. +-inverses 99.1%

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

      5. +-lft-identity 99.1%

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

   5. Simplified 99.1%

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

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 6.9%

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

      2. div-inv 6.9%

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

      3. add-sqr-sqrt 6.9%

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

      4. sub-neg 6.9%

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

      5. add-sqr-sqrt 3.1%

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

      6. hypot-def 3.1%

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

   3. Applied egg-rr 3.1%

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

      1. associate-*r/ 3.1%

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

      2. *-rgt-identity 3.1%

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

      3. associate--r- 60.6%

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

      4. +-inverses 60.6%

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

      5. +-lft-identity 60.6%

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

   5. Simplified 60.6%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. unpow2 0.0%

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

   8. Simplified 92.3%

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

      1. fma-udef 92.3%

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

   10. Applied egg-rr 92.3%

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

       1. unpow3 92.3%

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

       2. times-frac 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2\right)}\\ \end{array} \]

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-152)
     t_0
     (/
      eps
      (fma
       -0.125
       (* (/ eps (* x x)) (/ eps x))
       (+ (* (/ eps x) -0.5) (* x 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = eps / fma(-0.125, ((eps / (x * x)) * (eps / x)), (((eps / x) * -0.5) + (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-152)
		tmp = t_0;
	else
		tmp = Float64(eps / fma(-0.125, Float64(Float64(eps / Float64(x * x)) * Float64(eps / x)), Float64(Float64(Float64(eps / x) * -0.5) + Float64(x * 2.0))));
	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-152], t$95$0, N[(eps / N[(-0.125 * N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps / x), $MachinePrecision] * -0.5), $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))) < -1.00000000000000007e-152

   1. Initial program 98.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 6.9%

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

      2. div-inv 6.9%

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

      3. add-sqr-sqrt 6.9%

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

      4. sub-neg 6.9%

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

      5. add-sqr-sqrt 3.1%

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

      6. hypot-def 3.1%

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

   3. Applied egg-rr 3.1%

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

      1. associate-*r/ 3.1%

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

      2. *-rgt-identity 3.1%

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

      3. associate--r- 60.6%

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

      4. +-inverses 60.6%

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

      5. +-lft-identity 60.6%

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

   5. Simplified 60.6%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. unpow2 0.0%

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

   8. Simplified 92.3%

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

      1. fma-udef 92.3%

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

   10. Applied egg-rr 92.3%

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

       1. unpow3 92.3%

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

       2. times-frac 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2\right)}\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-152) t_0 (/ eps (+ (* (/ eps x) -0.5) (* x 2.0))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = eps / (((eps / x) * -0.5) + (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-152)) then
        tmp = t_0
    else
        tmp = eps / (((eps / x) * (-0.5d0)) + (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-152) {
		tmp = t_0;
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-152:
		tmp = t_0
	else:
		tmp = eps / (((eps / x) * -0.5) + (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-152)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(Float64(eps / x) * -0.5) + 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-152)
		tmp = t_0;
	else
		tmp = eps / (((eps / x) * -0.5) + (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-152], t$95$0, N[(eps / N[(N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000007e-152

   1. Initial program 98.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 6.9%

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

      2. div-inv 6.9%

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

      3. add-sqr-sqrt 6.9%

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

      4. sub-neg 6.9%

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

      5. add-sqr-sqrt 3.1%

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

      6. hypot-def 3.1%

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

   3. Applied egg-rr 3.1%

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

      1. associate-*r/ 3.1%

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

      2. *-rgt-identity 3.1%

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

      3. associate--r- 60.6%

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

      4. +-inverses 60.6%

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

      5. +-lft-identity 60.6%

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

   5. Simplified 60.6%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 99.7%

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

      5. associate-*r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. associate-*r/ 99.7%

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

      8. *-commutative 99.7%

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

      9. fma-def 99.7%

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

      10. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. fma-udef 92.3%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-93}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.55e-93)
   (- x (sqrt (- eps)))
   (/ eps (+ (* (/ eps x) -0.5) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.55e-93) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.55d-93) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (((eps / x) * (-0.5d0)) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.55e-93) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.55e-93:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.55e-93)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(Float64(eps / x) * -0.5) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.55e-93)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.55e-93], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55e-93

   1. Initial program 92.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around 0 92.3%

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

      1. neg-mul-1 92.3%

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

   4. Simplified 92.3%

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

   if 1.55e-93 < x

   1. Initial program 20.5%

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

      1. flip-- 20.6%

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

      2. div-inv 20.5%

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

      3. add-sqr-sqrt 20.6%

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

      4. sub-neg 20.6%

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

      5. add-sqr-sqrt 17.2%

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

      6. hypot-def 17.2%

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

   3. Applied egg-rr 17.2%

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

      1. associate-*r/ 17.2%

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

      2. *-rgt-identity 17.2%

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

      3. associate--r- 66.6%

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

      4. +-inverses 66.6%

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

      5. +-lft-identity 66.6%

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

   5. Simplified 66.6%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 88.1%

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

      5. associate-*r* 88.1%

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

      6. metadata-eval 88.1%

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

      7. associate-*r/ 88.1%

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

      8. *-commutative 88.1%

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

      9. fma-def 88.1%

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

      10. *-commutative 88.1%

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

   8. Simplified 88.1%

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

      1. fma-udef 87.7%

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

   10. Applied egg-rr 88.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-93}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 5: 45.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

   1. flip-- 61.4%

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

   2. div-inv 61.2%

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

   3. add-sqr-sqrt 61.0%

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

   4. sub-neg 61.0%

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

   5. add-sqr-sqrt 59.5%

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

   6. hypot-def 59.5%

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

3. Applied egg-rr 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-rgt-identity 59.5%

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

   3. associate--r- 83.5%

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

   4. +-inverses 83.5%

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

   5. +-lft-identity 83.5%

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

5. Simplified 83.5%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 45.5%

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

   5. associate-*r* 45.5%

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

   6. metadata-eval 45.5%

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

   7. associate-*r/ 45.5%

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

   8. *-commutative 45.5%

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

   9. fma-def 45.5%

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

   10. *-commutative 45.5%

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

8. Simplified 45.5%

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

   1. fma-udef 40.2%

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

10. Applied egg-rr 45.5%

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

11. Final simplification 45.5%

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

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 61.5%

   \[x - \sqrt{x \cdot x - \varepsilon} \]

2. Taylor expanded in x around inf 44.8%

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

3. Final simplification 44.8%

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

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 61.5%

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

   1. flip-- 61.4%

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

   2. div-inv 61.2%

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

   3. add-sqr-sqrt 61.0%

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

   4. sub-neg 61.0%

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

   5. add-sqr-sqrt 59.5%

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

   6. hypot-def 59.5%

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

3. Applied egg-rr 59.5%

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

   1. associate-*r/ 59.5%

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

   2. *-rgt-identity 59.5%

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

   3. associate--r- 83.5%

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

   4. +-inverses 83.5%

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

   5. +-lft-identity 83.5%

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

5. Simplified 83.5%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 45.5%

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

   5. associate-*r* 45.5%

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

   6. metadata-eval 45.5%

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

   7. associate-*r/ 45.5%

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

   8. *-commutative 45.5%

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

   9. fma-def 45.5%

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

   10. *-commutative 45.5%

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

8. Simplified 45.5%

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

9. Taylor expanded in eps around inf 5.6%

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

    1. *-commutative 5.6%

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

11. Simplified 5.6%

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

12. Final simplification 5.6%

    \[\leadsto x \cdot -2 \]

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 61.5%

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

   1. add-sqr-sqrt 61.0%

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

   2. pow2 61.0%

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

   3. pow1/2 61.0%

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

   4. sqrt-pow1 60.9%

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

   5. metadata-eval 60.9%

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

3. Applied egg-rr 60.9%

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

4. Taylor expanded in x around inf 4.3%

   \[\leadsto \color{blue}{x + -1 \cdot x} \]
5. 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%

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

6. Simplified 4.3%

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

7. Final simplification 4.3%

   \[\leadsto 0 \]

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 2023293 
(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))))