ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 99.5%

Time: 10.9s

Alternatives: 8

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x (sqrt (- (pow x 2.0) eps)))))

C

double code(double x, double eps) {
	return eps / (x + sqrt((pow(x, 2.0) - eps)));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return eps / (x + Math.sqrt((Math.pow(x, 2.0) - eps)));
}

Python

def code(x, eps):
	return eps / (x + math.sqrt((math.pow(x, 2.0) - eps)))

Julia

function code(x, eps)
	return Float64(eps / Float64(x + sqrt(Float64((x ^ 2.0) - eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (x + sqrt(((x ^ 2.0) - eps)));
end

Wolfram

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

Derivation

1. Initial program 64.9%

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

   1. add-sqr-sqrt 64.4%

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

   2. pow2 64.4%

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

   3. pow1/2 64.4%

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

   4. sqrt-pow1 64.4%

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

   5. pow2 64.4%

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

   6. metadata-eval 64.4%

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

4. Applied egg-rr 64.4%

   \[\leadsto x - \color{blue}{{\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]
5. Step-by-step derivation

   1. flip-- 64.4%

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

   2. div-inv 64.3%

      \[\leadsto \color{blue}{\left(x \cdot x - {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}}} \]

   3. unpow2 64.3%

      \[\leadsto \left(\color{blue}{{x}^{2}} - {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   4. pow-pow 64.6%

      \[\leadsto \left({x}^{2} - \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   5. metadata-eval 64.6%

      \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.5}} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   6. pow-pow 64.3%

      \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{0.5} \cdot \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   7. metadata-eval 64.3%

      \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{0.5} \cdot {\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.5}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   8. pow-prod-up 64.3%

      \[\leadsto \left({x}^{2} - \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.5 + 0.5\right)}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   9. metadata-eval 64.3%

      \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{\color{blue}{1}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   10. pow1 64.3%

       \[\leadsto \left({x}^{2} - \color{blue}{\left({x}^{2} - \varepsilon\right)}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   11. pow-pow 64.4%

       \[\leadsto \left({x}^{2} - \left({x}^{2} - \varepsilon\right)\right) \cdot \frac{1}{x + \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}}} \]

   12. metadata-eval 64.4%

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

   13. pow1/2 64.4%

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

6. Applied egg-rr 64.4%

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

   1. associate-*r/ 64.5%

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

   2. *-rgt-identity 64.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

   \[\leadsto \frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \]
10. Add Preprocessing

Alternative 2: 99.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-154)
   (- x (hypot (sqrt (- eps)) x))
   (/ eps (+ (* -0.5 (/ eps x)) (* x 2.0)))))

C

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

Java

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

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-154:
		tmp = x - math.hypot(math.sqrt(-eps), x)
	else:
		tmp = eps / ((-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))) <= -1e-154)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / Float64(Float64(-0.5 * Float64(eps / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154)
		tmp = x - hypot(sqrt(-eps), x);
	else
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	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-154], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $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))) < -9.9999999999999997e-155

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. hypot-define 99.5%

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

   4. Applied egg-rr 99.5%

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

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

   1. Initial program 10.9%

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

      1. flip-- 10.9%

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

      2. div-inv 10.9%

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

      3. add-sqr-sqrt 11.0%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 43.8%

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

      9. hypot-define 43.8%

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

   4. Applied egg-rr 43.8%

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

      1. *-commutative 43.8%

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

      2. +-inverses 43.8%

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

      3. +-lft-identity 43.8%

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

      4. associate-*l/ 44.0%

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

      5. *-lft-identity 44.0%

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

   6. Simplified 44.0%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 98.4%

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

      9. neg-mul-1 98.4%

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

      10. associate-*r/ 98.4%

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

      11. distribute-lft-neg-in 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. distribute-neg-frac 98.4%

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

      14. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in eps around 0 98.4%

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

4. Final simplification 99.1%

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

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154) t_0 (/ eps (+ (* -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 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-154:
		tmp = t_0
	else:
		tmp = eps / ((-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 <= -1e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(-0.5 * Float64(eps / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-154], t$95$0, N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $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))) < -9.9999999999999997e-155

   1. Initial program 99.5%

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

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

   1. Initial program 10.9%

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

      1. flip-- 10.9%

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

      2. div-inv 10.9%

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

      3. add-sqr-sqrt 11.0%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 43.8%

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

      9. hypot-define 43.8%

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

   4. Applied egg-rr 43.8%

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

      1. *-commutative 43.8%

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

      2. +-inverses 43.8%

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

      3. +-lft-identity 43.8%

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

      4. associate-*l/ 44.0%

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

      5. *-lft-identity 44.0%

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

   6. Simplified 44.0%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 98.4%

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

      9. neg-mul-1 98.4%

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

      10. associate-*r/ 98.4%

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

      11. distribute-lft-neg-in 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. distribute-neg-frac 98.4%

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

      14. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in eps around 0 98.4%

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

4. Final simplification 99.1%

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

Alternative 4: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\varepsilon}\\ \mathbf{if}\;x \leq 9.2 \cdot 10^{-122}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-99} \lor \neg \left(x \leq 1.55 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sqrt (- eps))))
   (if (<= x 9.2e-122)
     (- x t_0)
     (if (or (<= x 9.5e-99) (not (<= x 1.55e-87)))
       (/ eps (+ (* -0.5 (/ eps x)) (* x 2.0)))
       (/ eps (+ x t_0))))))

C

double code(double x, double eps) {
	double t_0 = sqrt(-eps);
	double tmp;
	if (x <= 9.2e-122) {
		tmp = x - t_0;
	} else if ((x <= 9.5e-99) || !(x <= 1.55e-87)) {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	} else {
		tmp = eps / (x + t_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 = sqrt(-eps)
    if (x <= 9.2d-122) then
        tmp = x - t_0
    else if ((x <= 9.5d-99) .or. (.not. (x <= 1.55d-87))) then
        tmp = eps / (((-0.5d0) * (eps / x)) + (x * 2.0d0))
    else
        tmp = eps / (x + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sqrt(-eps);
	double tmp;
	if (x <= 9.2e-122) {
		tmp = x - t_0;
	} else if ((x <= 9.5e-99) || !(x <= 1.55e-87)) {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	} else {
		tmp = eps / (x + t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sqrt(-eps)
	tmp = 0
	if x <= 9.2e-122:
		tmp = x - t_0
	elif (x <= 9.5e-99) or not (x <= 1.55e-87):
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0))
	else:
		tmp = eps / (x + t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = sqrt(Float64(-eps))
	tmp = 0.0
	if (x <= 9.2e-122)
		tmp = Float64(x - t_0);
	elseif ((x <= 9.5e-99) || !(x <= 1.55e-87))
		tmp = Float64(eps / Float64(Float64(-0.5 * Float64(eps / x)) + Float64(x * 2.0)));
	else
		tmp = Float64(eps / Float64(x + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sqrt(-eps);
	tmp = 0.0;
	if (x <= 9.2e-122)
		tmp = x - t_0;
	elseif ((x <= 9.5e-99) || ~((x <= 1.55e-87)))
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	else
		tmp = eps / (x + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sqrt[(-eps)], $MachinePrecision]}, If[LessEqual[x, 9.2e-122], N[(x - t$95$0), $MachinePrecision], If[Or[LessEqual[x, 9.5e-99], N[Not[LessEqual[x, 1.55e-87]], $MachinePrecision]], N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.20000000000000028e-122

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 97.5%

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

      1. neg-mul-1 97.5%

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

   5. Simplified 97.5%

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

   if 9.20000000000000028e-122 < x < 9.5000000000000008e-99 or 1.54999999999999999e-87 < x

   1. Initial program 31.1%

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

      1. flip-- 31.1%

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

      2. div-inv 31.0%

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

      3. add-sqr-sqrt 31.0%

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

      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 56.6%

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

      9. hypot-define 56.6%

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

   4. Applied egg-rr 56.6%

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

      1. *-commutative 56.6%

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

      2. +-inverses 56.6%

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

      3. +-lft-identity 56.6%

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

      4. associate-*l/ 56.8%

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

      5. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 78.7%

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

      9. neg-mul-1 78.7%

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

      10. associate-*r/ 78.7%

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

      11. distribute-lft-neg-in 78.7%

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

      12. distribute-rgt-neg-in 78.7%

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

      13. distribute-neg-frac 78.7%

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

      14. metadata-eval 78.7%

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

   9. Simplified 78.7%

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

   10. Taylor expanded in eps around 0 78.7%

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

   if 9.5000000000000008e-99 < x < 1.54999999999999999e-87

   1. Initial program 73.9%

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

      1. add-sqr-sqrt 73.9%

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

      2. pow2 73.9%

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

      3. pow1/2 73.9%

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

      4. sqrt-pow1 73.9%

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

      5. pow2 73.9%

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

      6. metadata-eval 73.9%

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

   4. Applied egg-rr 73.9%

      \[\leadsto x - \color{blue}{{\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. flip-- 73.9%

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

      2. div-inv 74.0%

         \[\leadsto \color{blue}{\left(x \cdot x - {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}}} \]

      3. unpow2 74.0%

         \[\leadsto \left(\color{blue}{{x}^{2}} - {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      4. pow-pow 74.2%

         \[\leadsto \left({x}^{2} - \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      5. metadata-eval 74.2%

         \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.5}} \cdot {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      6. pow-pow 73.8%

         \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{0.5} \cdot \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      7. metadata-eval 73.8%

         \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{0.5} \cdot {\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.5}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      8. pow-prod-up 73.1%

         \[\leadsto \left({x}^{2} - \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.5 + 0.5\right)}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      9. metadata-eval 73.1%

         \[\leadsto \left({x}^{2} - {\left({x}^{2} - \varepsilon\right)}^{\color{blue}{1}}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      10. pow1 73.1%

          \[\leadsto \left({x}^{2} - \color{blue}{\left({x}^{2} - \varepsilon\right)}\right) \cdot \frac{1}{x + {\left({\left({x}^{2} - \varepsilon\right)}^{0.25}\right)}^{2}} \]

      11. pow-pow 73.1%

          \[\leadsto \left({x}^{2} - \left({x}^{2} - \varepsilon\right)\right) \cdot \frac{1}{x + \color{blue}{{\left({x}^{2} - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}}} \]

      12. metadata-eval 73.1%

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

      13. pow1/2 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-*r/ 73.2%

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

      2. *-rgt-identity 73.2%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 77.1%

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

       1. neg-mul-1 77.1%

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

   11. Simplified 77.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-122}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-99} \lor \neg \left(x \leq 1.55 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \sqrt{-\varepsilon}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-122} \lor \neg \left(x \leq 2.15 \cdot 10^{-92}\right) \land x \leq 2.1 \cdot 10^{-87}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x 9.2e-122) (and (not (<= x 2.15e-92)) (<= x 2.1e-87)))
   (- x (sqrt (- eps)))
   (/ eps (+ (* -0.5 (/ eps x)) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= 9.2e-122) || (!(x <= 2.15e-92) && (x <= 2.1e-87))) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (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 <= 9.2d-122) .or. (.not. (x <= 2.15d-92)) .and. (x <= 2.1d-87)) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (((-0.5d0) * (eps / x)) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= 9.2e-122) || (!(x <= 2.15e-92) && (x <= 2.1e-87))) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= 9.2e-122) or (not (x <= 2.15e-92) and (x <= 2.1e-87)):
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= 9.2e-122) || (!(x <= 2.15e-92) && (x <= 2.1e-87)))
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(-0.5 * Float64(eps / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 9.2e-122) || (~((x <= 2.15e-92)) && (x <= 2.1e-87)))
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, 9.2e-122], And[N[Not[LessEqual[x, 2.15e-92]], $MachinePrecision], LessEqual[x, 2.1e-87]]], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.20000000000000028e-122 or 2.15000000000000007e-92 < x < 2.10000000000000007e-87

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0 96.9%

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

      1. neg-mul-1 96.9%

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

   5. Simplified 96.9%

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

   if 9.20000000000000028e-122 < x < 2.15000000000000007e-92 or 2.10000000000000007e-87 < x

   1. Initial program 31.8%

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

      1. flip-- 31.7%

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

      2. div-inv 31.6%

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

      3. add-sqr-sqrt 31.6%

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

      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 57.9%

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

      9. hypot-define 57.9%

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

   4. Applied egg-rr 57.9%

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

      1. *-commutative 57.9%

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

      2. +-inverses 57.9%

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

      3. +-lft-identity 57.9%

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

      4. associate-*l/ 58.1%

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

      5. *-lft-identity 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 77.9%

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

      9. neg-mul-1 77.9%

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

      10. associate-*r/ 77.9%

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

      11. distribute-lft-neg-in 77.9%

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

      12. distribute-rgt-neg-in 77.9%

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

      13. distribute-neg-frac 77.9%

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

      14. metadata-eval 77.9%

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

   9. Simplified 77.9%

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

   10. Taylor expanded in eps around 0 77.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-122} \lor \neg \left(x \leq 2.15 \cdot 10^{-92}\right) \land x \leq 2.1 \cdot 10^{-87}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.2% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

   1. flip-- 64.8%

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

   2. div-inv 64.6%

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

   3. add-sqr-sqrt 64.4%

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

   4. associate--r- 99.3%

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

   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 77.5%

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

   9. hypot-define 77.5%

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

4. Applied egg-rr 77.5%

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

   1. *-commutative 77.5%

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

   2. +-inverses 77.5%

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

   3. +-lft-identity 77.5%

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

   4. associate-*l/ 77.7%

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

   5. *-lft-identity 77.7%

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

6. Simplified 77.7%

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

7. Taylor expanded in eps around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. *-commutative 0.0%

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

   3. fma-define 0.0%

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

   4. associate-*r/ 0.0%

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

   5. *-commutative 0.0%

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

   6. *-commutative 0.0%

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

   7. unpow2 0.0%

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

   8. rem-square-sqrt 42.9%

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

   9. neg-mul-1 42.9%

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

   10. associate-*r/ 42.9%

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

   11. distribute-lft-neg-in 42.9%

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

   12. distribute-rgt-neg-in 42.9%

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

   13. distribute-neg-frac 42.9%

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

   14. metadata-eval 42.9%

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

9. Simplified 42.9%

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

10. Taylor expanded in eps around 0 42.9%

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

11. Final simplification 42.9%

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

Alternative 7: 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 64.9%

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

3. Taylor expanded in x around inf 41.7%

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

4. Final simplification 41.7%

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

Alternative 8: 5.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* x -2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

   1. flip-- 64.8%

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

   2. div-inv 64.6%

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

   3. add-sqr-sqrt 64.4%

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

   4. associate--r- 99.3%

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

   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 77.5%

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

   9. hypot-define 77.5%

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

4. Applied egg-rr 77.5%

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

   1. *-commutative 77.5%

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

   2. +-inverses 77.5%

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

   3. +-lft-identity 77.5%

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

   4. associate-*l/ 77.7%

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

   5. *-lft-identity 77.7%

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

6. Simplified 77.7%

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

7. Taylor expanded in eps around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. *-commutative 0.0%

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

   3. fma-define 0.0%

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

   4. associate-*r/ 0.0%

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

   5. *-commutative 0.0%

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

   6. *-commutative 0.0%

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

   7. unpow2 0.0%

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

   8. rem-square-sqrt 42.9%

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

   9. neg-mul-1 42.9%

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

   10. associate-*r/ 42.9%

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

   11. distribute-lft-neg-in 42.9%

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

   12. distribute-rgt-neg-in 42.9%

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

   13. distribute-neg-frac 42.9%

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

   14. metadata-eval 42.9%

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

9. Simplified 42.9%

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

10. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

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

12. Simplified 5.3%

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

13. Final simplification 5.3%

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

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 2024079 
(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)))

  :alt
  (/ eps (+ x (sqrt (- (* x x) eps))))

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