ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.8% → 99.1%

Time: 5.4s

Alternatives: 11

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: 61.8% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \left(x + x\right)\right) + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-153)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/
    eps
    (+ (+ (/ -0.125 (/ x (pow (/ eps x) 2.0))) (+ x x)) (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / (((-0.125 / (x / pow((eps / x), 2.0))) + (x + x)) + ((eps / x) * -0.5));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / (((-0.125 / (x / Math.pow((eps / x), 2.0))) + (x + x)) + ((eps / x) * -0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-153:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / (((-0.125 / (x / math.pow((eps / x), 2.0))) + (x + x)) + ((eps / x) * -0.5))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-153)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(Float64(Float64(-0.125 / Float64(x / (Float64(eps / x) ^ 2.0))) + Float64(x + x)) + Float64(Float64(eps / x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / (((-0.125 / (x / ((eps / x) ^ 2.0))) + (x + x)) + ((eps / x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-153], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(N[(-0.125 / N[(x / N[Power[N[(eps / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.7%

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

      1. flip-- 98.8%

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

      2. div-inv 98.4%

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

      3. add-sqr-sqrt 98.2%

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

      4. sub-neg 98.2%

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

      5. add-sqr-sqrt 98.2%

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

      6. hypot-def 98.2%

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

   3. Applied egg-rr 98.2%

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

      1. associate-*r/ 98.1%

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

      2. *-rgt-identity 98.1%

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

      3. associate--r- 99.2%

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

      4. +-inverses 99.2%

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

      5. +-lft-identity 99.2%

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

   5. Simplified 99.2%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.2%

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

      1. flip-- 8.2%

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

      2. div-inv 8.2%

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

      3. add-sqr-sqrt 8.3%

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

      4. sub-neg 8.3%

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

      5. add-sqr-sqrt 2.5%

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

      6. hypot-def 2.5%

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. associate--r- 43.5%

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

      4. +-inverses 43.5%

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

      5. +-lft-identity 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 0.0%

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

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

      3. pow-sqr 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 0.0%

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

      8. metadata-eval 0.0%

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

      9. *-rgt-identity 0.0%

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

      10. unpow2 0.0%

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

      11. +-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. fma-def 0.0%

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

      14. associate-*r/ 0.0%

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

   8. Simplified 93.0%

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

      1. fma-udef 93.0%

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

      2. fma-udef 93.0%

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

      3. *-commutative 93.0%

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

      4. count-2 93.0%

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

      5. associate-+r+ 93.0%

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

      6. clear-num 93.0%

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

      7. un-div-inv 93.0%

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

      8. cube-mult 93.0%

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

      9. associate-/l* 100.0%

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

      10. frac-times 100.0%

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

      11. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \left(x + x\right)\right) + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \]

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \left(x + x\right)\right) + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-153)
   (- x (hypot (sqrt (- eps)) x))
   (/
    eps
    (+ (+ (/ -0.125 (/ x (pow (/ eps x) 2.0))) (+ x x)) (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / (((-0.125 / (x / pow((eps / x), 2.0))) + (x + x)) + ((eps / x) * -0.5));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = x - Math.hypot(Math.sqrt(-eps), x);
	} else {
		tmp = eps / (((-0.125 / (x / Math.pow((eps / x), 2.0))) + (x + x)) + ((eps / x) * -0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-153:
		tmp = x - math.hypot(math.sqrt(-eps), x)
	else:
		tmp = eps / (((-0.125 / (x / math.pow((eps / x), 2.0))) + (x + x)) + ((eps / x) * -0.5))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-153)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / Float64(Float64(Float64(-0.125 / Float64(x / (Float64(eps / x) ^ 2.0))) + Float64(x + x)) + Float64(Float64(eps / x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153)
		tmp = x - hypot(sqrt(-eps), x);
	else
		tmp = eps / (((-0.125 / (x / ((eps / x) ^ 2.0))) + (x + x)) + ((eps / x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-153], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(N[(-0.125 / N[(x / N[Power[N[(eps / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.7%

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

      1. sub-neg 98.7%

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

      2. +-commutative 98.7%

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

      3. add-sqr-sqrt 98.8%

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

      4. hypot-def 98.8%

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

   3. Applied egg-rr 98.8%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.2%

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

      1. flip-- 8.2%

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

      2. div-inv 8.2%

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

      3. add-sqr-sqrt 8.3%

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

      4. sub-neg 8.3%

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

      5. add-sqr-sqrt 2.5%

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

      6. hypot-def 2.5%

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. associate--r- 43.5%

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

      4. +-inverses 43.5%

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

      5. +-lft-identity 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 0.0%

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

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

      3. pow-sqr 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 0.0%

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

      8. metadata-eval 0.0%

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

      9. *-rgt-identity 0.0%

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

      10. unpow2 0.0%

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

      11. +-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. fma-def 0.0%

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

      14. associate-*r/ 0.0%

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

   8. Simplified 93.0%

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

      1. fma-udef 93.0%

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

      2. fma-udef 93.0%

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

      3. *-commutative 93.0%

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

      4. count-2 93.0%

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

      5. associate-+r+ 93.0%

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

      6. clear-num 93.0%

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

      7. un-div-inv 93.0%

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

      8. cube-mult 93.0%

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

      9. associate-/l* 100.0%

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

      10. frac-times 100.0%

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

      11. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \left(x + x\right)\right) + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \]

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \left(x + x\right)\right) + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-153)
     t_0
     (/
      eps
      (+
       (+ (/ -0.125 (/ x (pow (/ eps x) 2.0))) (+ 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-153) {
		tmp = t_0;
	} else {
		tmp = eps / (((-0.125 / (x / pow((eps / x), 2.0))) + (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-153)) then
        tmp = t_0
    else
        tmp = eps / ((((-0.125d0) / (x / ((eps / x) ** 2.0d0))) + (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-153) {
		tmp = t_0;
	} else {
		tmp = eps / (((-0.125 / (x / Math.pow((eps / x), 2.0))) + (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-153:
		tmp = t_0
	else:
		tmp = eps / (((-0.125 / (x / math.pow((eps / x), 2.0))) + (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-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(Float64(-0.125 / Float64(x / (Float64(eps / x) ^ 2.0))) + Float64(x + 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-153)
		tmp = t_0;
	else
		tmp = eps / (((-0.125 / (x / ((eps / x) ^ 2.0))) + (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-153], t$95$0, N[(eps / N[(N[(N[(-0.125 / N[(x / N[Power[N[(eps / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.7%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.2%

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

      1. flip-- 8.2%

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

      2. div-inv 8.2%

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

      3. add-sqr-sqrt 8.3%

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

      4. sub-neg 8.3%

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

      5. add-sqr-sqrt 2.5%

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

      6. hypot-def 2.5%

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. associate--r- 43.5%

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

      4. +-inverses 43.5%

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

      5. +-lft-identity 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 0.0%

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

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

      3. pow-sqr 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 0.0%

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

      8. metadata-eval 0.0%

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

      9. *-rgt-identity 0.0%

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

      10. unpow2 0.0%

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

      11. +-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. fma-def 0.0%

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

      14. associate-*r/ 0.0%

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

   8. Simplified 93.0%

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

      1. fma-udef 93.0%

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

      2. fma-udef 93.0%

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

      3. *-commutative 93.0%

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

      4. count-2 93.0%

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

      5. associate-+r+ 93.0%

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

      6. clear-num 93.0%

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

      7. un-div-inv 93.0%

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

      8. cube-mult 93.0%

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

      9. associate-/l* 100.0%

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

      10. frac-times 100.0%

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

      11. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\left(\frac{-0.125}{\frac{x}{{\left(\frac{\varepsilon}{x}\right)}^{2}}} + \left(x + x\right)\right) + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \]

Alternative 4: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\varepsilon}{x}\right)}^{2} \cdot \frac{0.125}{x} + \frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-153)
     t_0
     (+ (* (pow (/ eps x) 2.0) (/ 0.125 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-153) {
		tmp = t_0;
	} else {
		tmp = (pow((eps / x), 2.0) * (0.125 / 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-153)) then
        tmp = t_0
    else
        tmp = (((eps / x) ** 2.0d0) * (0.125d0 / 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-153) {
		tmp = t_0;
	} else {
		tmp = (Math.pow((eps / x), 2.0) * (0.125 / 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-153:
		tmp = t_0
	else:
		tmp = (math.pow((eps / x), 2.0) * (0.125 / 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-153)
		tmp = t_0;
	else
		tmp = Float64(Float64((Float64(eps / x) ^ 2.0) * Float64(0.125 / 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-153)
		tmp = t_0;
	else
		tmp = (((eps / x) ^ 2.0) * (0.125 / 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-153], t$95$0, N[(N[(N[Power[N[(eps / x), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.125 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.7%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.2%

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

      1. flip-- 8.2%

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

      2. div-inv 8.2%

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

      3. add-sqr-sqrt 8.3%

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

      4. sub-neg 8.3%

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

      5. add-sqr-sqrt 2.5%

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

      6. hypot-def 2.5%

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. associate--r- 43.5%

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

      4. +-inverses 43.5%

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

      5. +-lft-identity 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. associate-/l* 0.0%

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

      4. associate-*r/ 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 92.7%

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

      7. associate-/l* 92.7%

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

      8. *-commutative 92.7%

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

      9. *-commutative 92.7%

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

      10. associate-*l* 92.7%

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

      11. *-commutative 92.7%

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

      12. associate-*l* 92.7%

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

      13. metadata-eval 92.7%

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

      14. /-rgt-identity 92.7%

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

      15. metadata-eval 92.7%

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

      16. rem-square-sqrt 0.0%

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

      17. unpow2 0.0%

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

      18. rem-square-sqrt 0.0%

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

      19. unpow2 0.0%

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

      20. pow-sqr 0.0%

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

      21. metadata-eval 0.0%

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

   8. Simplified 92.7%

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

      1. fma-udef 92.7%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x} + \frac{\varepsilon \cdot \varepsilon}{{x}^{3}} \cdot 0.125} \]

      2. +-commutative 92.7%

         \[\leadsto \color{blue}{\frac{\varepsilon \cdot \varepsilon}{{x}^{3}} \cdot 0.125 + 0.5 \cdot \frac{\varepsilon}{x}} \]

      3. associate-*l/ 92.7%

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

      4. unpow3 92.7%

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

      5. times-frac 99.5%

         \[\leadsto \color{blue}{\frac{\varepsilon \cdot \varepsilon}{x \cdot x} \cdot \frac{0.125}{x}} + 0.5 \cdot \frac{\varepsilon}{x} \]

      6. frac-times 99.8%

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

      7. pow2 99.8%

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

      8. *-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\varepsilon}{x}\right)}^{2} \cdot \frac{0.125}{x} + \frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]

Alternative 5: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-153) 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-153) {
		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-153)) 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-153) {
		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-153:
		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-153)
		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-153)
		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-153], 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))) < -1.00000000000000004e-153

   1. Initial program 98.7%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.2%

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

      1. flip-- 8.2%

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

      2. div-inv 8.2%

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

      3. add-sqr-sqrt 8.3%

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

      4. sub-neg 8.3%

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

      5. add-sqr-sqrt 2.5%

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

      6. hypot-def 2.5%

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. associate--r- 43.5%

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

      4. +-inverses 43.5%

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

      5. +-lft-identity 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 99.8%

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

      5. associate-*r* 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r/ 99.8%

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

      8. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 6: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 6.5e-100)
   (- x (sqrt (- eps)))
   (/ eps (+ (* (/ eps x) -0.5) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 6.5e-100) {
		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 <= 6.5d-100) 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 <= 6.5e-100) {
		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 <= 6.5e-100:
		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 <= 6.5e-100)
		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 <= 6.5e-100)
		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, 6.5e-100], 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 < 6.50000000000000013e-100

   1. Initial program 90.6%

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

   2. Taylor expanded in x around 0 89.7%

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

      1. neg-mul-1 89.7%

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

   4. Simplified 89.7%

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

   if 6.50000000000000013e-100 < x

   1. Initial program 26.0%

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

      1. flip-- 26.1%

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

      2. div-inv 26.0%

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

      3. add-sqr-sqrt 26.2%

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

      4. sub-neg 26.2%

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

      5. add-sqr-sqrt 21.4%

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

      6. hypot-def 21.4%

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

   3. Applied egg-rr 21.4%

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

      1. associate-*r/ 21.4%

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

      2. *-rgt-identity 21.4%

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

      3. associate--r- 54.1%

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

      4. +-inverses 54.1%

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

      5. +-lft-identity 54.1%

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

   5. Simplified 54.1%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 83.1%

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

      5. associate-*r* 83.1%

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

      6. metadata-eval 83.1%

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

      7. associate-*r/ 83.1%

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

      8. *-commutative 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in x around 0 83.1%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 7: 45.4% 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 58.1%

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

   1. flip-- 58.1%

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

   2. div-inv 57.9%

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

   3. add-sqr-sqrt 57.8%

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

   4. sub-neg 57.8%

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

   5. add-sqr-sqrt 55.2%

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

   6. hypot-def 55.2%

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

3. Applied egg-rr 55.2%

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

   1. associate-*r/ 55.2%

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

   2. *-rgt-identity 55.2%

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

   3. associate--r- 74.2%

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

   4. +-inverses 74.2%

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

   5. +-lft-identity 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 49.4%

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

   5. associate-*r* 49.4%

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

   6. metadata-eval 49.4%

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

   7. associate-*r/ 49.4%

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

   8. *-commutative 49.4%

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

8. Simplified 49.4%

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

9. Final simplification 49.4%

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

Alternative 8: 45.4% 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 58.1%

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

   1. flip-- 58.1%

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

   2. div-inv 57.9%

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

   3. add-sqr-sqrt 57.8%

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

   4. sub-neg 57.8%

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

   5. add-sqr-sqrt 55.2%

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

   6. hypot-def 55.2%

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

3. Applied egg-rr 55.2%

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

   1. associate-*r/ 55.2%

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

   2. *-rgt-identity 55.2%

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

   3. associate--r- 74.2%

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

   4. +-inverses 74.2%

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

   5. +-lft-identity 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 49.4%

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

   5. associate-*r* 49.4%

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

   6. metadata-eval 49.4%

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

   7. associate-*r/ 49.4%

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

   8. *-commutative 49.4%

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

8. Simplified 49.4%

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

9. Taylor expanded in x around 0 49.4%

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

10. Final simplification 49.4%

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

Alternative 9: 44.6% 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 58.1%

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

2. Taylor expanded in x around inf 48.6%

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

3. Final simplification 48.6%

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

Alternative 10: 44.6% accurate, 21.4× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / (x + x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

   1. flip-- 58.1%

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

   2. div-inv 57.9%

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

   3. add-sqr-sqrt 57.8%

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

   4. sub-neg 57.8%

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

   5. add-sqr-sqrt 55.2%

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

   6. hypot-def 55.2%

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

3. Applied egg-rr 55.2%

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

   1. associate-*r/ 55.2%

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

   2. *-rgt-identity 55.2%

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

   3. associate--r- 74.2%

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

   4. +-inverses 74.2%

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

   5. +-lft-identity 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in x around inf 48.6%

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

7. Final simplification 48.6%

   \[\leadsto \frac{\varepsilon}{x + x} \]

Alternative 11: 5.4% 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 58.1%

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

   1. flip-- 58.1%

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

   2. div-inv 57.9%

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

   3. add-sqr-sqrt 57.8%

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

   4. sub-neg 57.8%

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

   5. add-sqr-sqrt 55.2%

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

   6. hypot-def 55.2%

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

3. Applied egg-rr 55.2%

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

   1. associate-*r/ 55.2%

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

   2. *-rgt-identity 55.2%

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

   3. associate--r- 74.2%

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

   4. +-inverses 74.2%

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

   5. +-lft-identity 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 49.4%

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

   5. associate-*r* 49.4%

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

   6. metadata-eval 49.4%

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

   7. associate-*r/ 49.4%

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

   8. *-commutative 49.4%

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

8. Simplified 49.4%

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

9. Taylor expanded in eps around inf 5.2%

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

    1. *-commutative 5.2%

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

11. Simplified 5.2%

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

12. Final simplification 5.2%

    \[\leadsto x \cdot -2 \]

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 2023271 
(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))))