ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.2% → 99.1%

Time: 11.1s

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: 62.2% 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, 0.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (/
    1.0
    (/ (+ x (hypot x (sqrt (- eps)))) (+ eps (- (pow x 2.0) (pow x 2.0)))))
   (/ eps (fma 0.5 (/ (- eps) x) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = 1.0 / ((x + hypot(x, sqrt(-eps))) / (eps + (pow(x, 2.0) - pow(x, 2.0))));
	} else {
		tmp = eps / fma(0.5, (-eps / x), (x * 2.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(1.0 / Float64(Float64(x + hypot(x, sqrt(Float64(-eps)))) / Float64(eps + Float64((x ^ 2.0) - (x ^ 2.0)))));
	else
		tmp = Float64(eps / fma(0.5, Float64(Float64(-eps) / x), 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], -2e-154], N[(1.0 / N[(N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(0.5 * N[((-eps) / x), $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))) < -1.9999999999999999e-154

   1. Initial program 98.5%

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

      1. flip-- 98.3%

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

      2. clear-num 98.1%

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

      3. sub-neg 98.1%

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

      4. add-sqr-sqrt 98.1%

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

      5. hypot-def 98.1%

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

      6. add-sqr-sqrt 97.9%

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

      7. associate--r- 99.3%

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

      8. pow2 99.3%

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

      9. pow2 99.3%

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

   4. Applied egg-rr 99.3%

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

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

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 7.6%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 45.3%

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

      9. hypot-def 45.3%

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

   4. Applied egg-rr 45.3%

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

      1. associate-*r/ 45.5%

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

      2. +-inverses 45.5%

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

      3. +-lft-identity 45.5%

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

      4. *-rgt-identity 45.5%

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

   6. Simplified 45.5%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 98.9%

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

      5. neg-mul-1 98.9%

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

      6. *-commutative 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 99.1%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(0.5, \frac{-\varepsilon}{x}, x \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-149)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (fma 0.5 (/ (- eps) x) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-149) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / fma(0.5, (-eps / x), (x * 2.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-149)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / fma(0.5, Float64(Float64(-eps) / x), 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], -5e-149], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(0.5 * N[((-eps) / x), $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))) < -4.99999999999999968e-149

   1. Initial program 98.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing
   3. 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 98.4%

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

      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. associate--r- 99.3%

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

      5. pow2 99.3%

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

      6. pow2 99.3%

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

      7. sub-neg 99.3%

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

      8. add-sqr-sqrt 99.3%

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

      9. hypot-def 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. +-inverses 99.3%

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

      3. +-lft-identity 99.3%

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

      4. *-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

   if -4.99999999999999968e-149 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.0%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 45.8%

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

      9. hypot-def 45.8%

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

   4. Applied egg-rr 45.8%

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

      1. associate-*r/ 46.0%

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

      2. +-inverses 46.0%

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

      3. +-lft-identity 46.0%

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

      4. *-rgt-identity 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 98.9%

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

      5. neg-mul-1 98.9%

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

      6. *-commutative 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(0.5, \frac{-\varepsilon}{x}, x \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(0.5, \frac{-\varepsilon}{x}, x \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-149)
   (- x (hypot (sqrt (- eps)) x))
   (/ eps (fma 0.5 (/ (- eps) x) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-149) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / fma(0.5, (-eps / x), (x * 2.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-149)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / fma(0.5, Float64(Float64(-eps) / x), 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], -5e-149], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(0.5 * N[((-eps) / x), $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))) < -4.99999999999999968e-149

   1. Initial program 98.8%

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

      1. sub-neg 98.8%

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

      2. +-commutative 98.8%

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

      3. add-sqr-sqrt 98.8%

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

      4. hypot-def 98.9%

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

   4. Applied egg-rr 98.9%

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

   if -4.99999999999999968e-149 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.0%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 45.8%

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

      9. hypot-def 45.8%

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

   4. Applied egg-rr 45.8%

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

      1. associate-*r/ 46.0%

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

      2. +-inverses 46.0%

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

      3. +-lft-identity 46.0%

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

      4. *-rgt-identity 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 98.9%

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

      5. neg-mul-1 98.9%

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

      6. *-commutative 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(0.5, \frac{-\varepsilon}{x}, x \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(0.5, \frac{-\varepsilon}{x}, x \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-149) t_0 (/ eps (fma 0.5 (/ (- eps) x) (* x 2.0))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-149) {
		tmp = t_0;
	} else {
		tmp = eps / fma(0.5, (-eps / x), (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 <= -5e-149)
		tmp = t_0;
	else
		tmp = Float64(eps / fma(0.5, Float64(Float64(-eps) / x), 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, -5e-149], t$95$0, N[(eps / N[(0.5 * N[((-eps) / x), $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))) < -4.99999999999999968e-149

   1. Initial program 98.8%

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

   if -4.99999999999999968e-149 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.0%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 45.8%

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

      9. hypot-def 45.8%

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

   4. Applied egg-rr 45.8%

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

      1. associate-*r/ 46.0%

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

      2. +-inverses 46.0%

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

      3. +-lft-identity 46.0%

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

      4. *-rgt-identity 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 98.9%

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

      5. neg-mul-1 98.9%

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

      6. *-commutative 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(0.5, \frac{-\varepsilon}{x}, x \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-149) t_0 (/ 1.0 (fma 2.0 (/ x eps) (/ -0.5 x))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-149) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(2.0, (x / eps), (-0.5 / x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-149)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(2.0, Float64(x / eps), Float64(-0.5 / x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-149], t$95$0, N[(1.0 / N[(2.0 * N[(x / eps), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.99999999999999968e-149

   1. Initial program 98.8%

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

   if -4.99999999999999968e-149 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

      1. flip-- 7.8%

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

      2. clear-num 7.8%

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

      3. sub-neg 7.8%

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

      4. add-sqr-sqrt 2.7%

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

      5. hypot-def 2.7%

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

      6. add-sqr-sqrt 2.7%

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

      7. associate--r- 45.8%

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

      8. pow2 45.8%

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

      9. pow2 45.8%

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

   4. Applied egg-rr 45.8%

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

   5. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 98.4%

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

      6. metadata-eval 98.4%

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

   7. Simplified 98.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-149) t_0 (/ (* eps 0.5) x))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-149) {
		tmp = t_0;
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-5d-149)) then
        tmp = t_0
    else
        tmp = (eps * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-149) {
		tmp = t_0;
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-149:
		tmp = t_0
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-149)
		tmp = t_0;
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -5e-149)
		tmp = t_0;
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-149], t$95$0, N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.99999999999999968e-149

   1. Initial program 98.8%

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

   if -4.99999999999999968e-149 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

   3. Taylor expanded in x around inf 98.0%

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

      1. associate-*r/ 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-128}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.4e-128) (- x (sqrt (- eps))) (/ (* eps 0.5) x)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.4e-128) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 3.4d-128) then
        tmp = x - sqrt(-eps)
    else
        tmp = (eps * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.4e-128) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.4e-128:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.4e-128)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.4e-128)
		tmp = x - sqrt(-eps);
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.4e-128], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999975e-128

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. neg-mul-1 97.7%

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

   5. Simplified 97.7%

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

   if 3.39999999999999975e-128 < x

   1. Initial program 29.5%

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

   3. Taylor expanded in x around inf 77.4%

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

      1. associate-*r/ 77.4%

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

   5. Simplified 77.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-128}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.0% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

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

3. Taylor expanded in x around inf 47.6%

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

   1. associate-*r/ 47.6%

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

   2. associate-/l* 47.4%

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

5. Simplified 47.4%

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

   1. associate-/r/ 47.4%

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

7. Applied egg-rr 47.4%

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

8. Final simplification 47.4%

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

Alternative 9: 44.1% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

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

3. Taylor expanded in x around inf 47.6%

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

   1. associate-*r/ 47.6%

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

5. Simplified 47.6%

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

6. Final simplification 47.6%

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

Alternative 10: 11.4% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

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

   1. flip-- 58.5%

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

   2. div-inv 58.4%

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

   3. add-sqr-sqrt 58.4%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 75.7%

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

   9. hypot-def 75.7%

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

4. Applied egg-rr 75.7%

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

   1. associate-*r/ 75.8%

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

   2. +-inverses 75.8%

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

   3. +-lft-identity 75.8%

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

   4. *-rgt-identity 75.8%

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

6. Simplified 75.8%

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

   1. add-sqr-sqrt 75.7%

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

   2. pow2 75.7%

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

8. Applied egg-rr 75.7%

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

9. Taylor expanded in x around inf 11.9%

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

10. Final simplification 11.9%

    \[\leadsto \frac{\varepsilon}{x} \]
11. Add Preprocessing

Alternative 11: 5.7% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 -1.0)

C

double code(double x, double eps) {
	return -1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = -1.0d0
end function

Java

public static double code(double x, double eps) {
	return -1.0;
}

Python

def code(x, eps):
	return -1.0

Julia

function code(x, eps)
	return -1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = -1.0;
end

Wolfram

code[x_, eps_] := -1.0

Derivation

1. Initial program 58.7%

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

   1. flip-- 58.5%

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

   2. div-inv 58.4%

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

   3. add-sqr-sqrt 58.4%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 75.7%

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

   9. hypot-def 75.7%

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

4. Applied egg-rr 75.7%

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

   1. associate-*r/ 75.8%

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

   2. +-inverses 75.8%

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

   3. +-lft-identity 75.8%

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

   4. *-rgt-identity 75.8%

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

6. Simplified 75.8%

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

   1. add-sqr-sqrt 75.7%

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

   2. pow2 75.7%

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

8. Applied egg-rr 75.7%

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

9. Taylor expanded in eps around 0 47.6%

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

11. Simplified 5.8%

    \[\leadsto \color{blue}{-1} \]

12. Final simplification 5.8%

    \[\leadsto -1 \]
13. Add Preprocessing

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024101 
(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))))