ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 98.5%

Time: 10.7s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153) t_0 (/ eps (+ x (+ x (* -0.5 (/ eps x))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / 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-153)) then
        tmp = t_0
    else
        tmp = eps / (x + (x + ((-0.5d0) * (eps / 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-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-153:
		tmp = t_0
	else:
		tmp = eps / (x + (x + (-0.5 * (eps / 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-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(-0.5 * Float64(eps / 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-153)
		tmp = t_0;
	else
		tmp = eps / (x + (x + (-0.5 * (eps / 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-153], t$95$0, N[(eps / N[(x + N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 100.0%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.1%

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

      1. add-sqr-sqrt 6.3%

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

      2. add-sqr-sqrt 6.9%

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

      3. difference-of-squares 7.0%

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

      4. pow1/2 7.0%

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

      5. sqrt-pow1 7.0%

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

      6. pow2 7.0%

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

      7. metadata-eval 7.0%

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

      8. pow1/2 7.0%

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

      9. sqrt-pow1 6.8%

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

      10. pow2 6.8%

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

      11. metadata-eval 6.8%

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

   4. Applied egg-rr 6.8%

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

      1. difference-of-squares 6.8%

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

      2. add-sqr-sqrt 6.7%

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

      3. flip-- 6.7%

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

      4. unpow2 6.7%

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

      5. pow-prod-up 6.9%

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

      6. metadata-eval 6.9%

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

      7. pow1/2 6.9%

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

      8. pow-prod-up 7.1%

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

      9. metadata-eval 7.1%

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

      10. pow1/2 7.1%

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

      11. add-sqr-sqrt 7.2%

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

      12. pow-prod-up 7.2%

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

   6. Applied egg-rr 7.2%

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

      1. associate--r- 100.0%

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

      2. +-inverses 100.0%

         \[\leadsto \frac{\color{blue}{0} + \varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \]

      3. +-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around 0 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 54.6%

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

   1. add-sqr-sqrt 54.2%

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

   2. add-sqr-sqrt 54.2%

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

   3. difference-of-squares 54.2%

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

   4. pow1/2 54.2%

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

   5. sqrt-pow1 54.3%

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

   6. pow2 54.3%

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

   7. metadata-eval 54.3%

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

   8. pow1/2 54.3%

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

   9. sqrt-pow1 54.1%

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

   10. pow2 54.1%

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

   11. metadata-eval 54.1%

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

4. Applied egg-rr 54.1%

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

   1. difference-of-squares 54.1%

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

   2. add-sqr-sqrt 54.0%

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

   3. flip-- 54.0%

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

   4. unpow2 54.0%

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

   5. pow-prod-up 54.5%

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

   6. metadata-eval 54.5%

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

   7. pow1/2 54.5%

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

   8. pow-prod-up 54.2%

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

   9. metadata-eval 54.2%

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

   10. pow1/2 54.2%

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

   11. add-sqr-sqrt 54.2%

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

   12. pow-prod-up 54.3%

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

6. Applied egg-rr 54.3%

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

   1. associate--r- 99.6%

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

   2. +-inverses 99.6%

      \[\leadsto \frac{\color{blue}{0} + \varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \]

   3. +-lft-identity 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.65e-108)
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.65e-108) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / 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 <= 1.65d-108) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + ((-0.5d0) * (eps / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.65e-108) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.65e-108:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + (-0.5 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.65e-108)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(-0.5 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.65e-108)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.65e-108], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6500000000000001e-108

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 92.9%

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

      1. neg-mul-1 92.9%

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

   5. Simplified 92.9%

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

   if 1.6500000000000001e-108 < x

   1. Initial program 20.9%

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

      1. add-sqr-sqrt 20.2%

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

      2. add-sqr-sqrt 20.6%

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

      3. difference-of-squares 20.6%

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

      4. pow1/2 20.6%

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

      5. sqrt-pow1 20.7%

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

      6. pow2 20.7%

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

      7. metadata-eval 20.7%

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

      8. pow1/2 20.7%

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

      9. sqrt-pow1 20.5%

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

      10. pow2 20.5%

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

      11. metadata-eval 20.5%

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

   4. Applied egg-rr 20.5%

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

      1. difference-of-squares 20.4%

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

      2. add-sqr-sqrt 20.3%

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

      3. flip-- 20.2%

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

      4. unpow2 20.2%

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

      5. pow-prod-up 20.6%

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

      6. metadata-eval 20.6%

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

      7. pow1/2 20.6%

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

      8. pow-prod-up 20.7%

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

      9. metadata-eval 20.7%

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

      10. pow1/2 20.7%

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

      11. add-sqr-sqrt 20.8%

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

      12. pow-prod-up 20.8%

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

   6. Applied egg-rr 20.8%

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

      1. associate--r- 99.9%

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

      2. +-inverses 99.9%

         \[\leadsto \frac{\color{blue}{0} + \varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \]

      3. +-lft-identity 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around 0 85.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. add-sqr-sqrt 54.2%

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

   2. add-sqr-sqrt 54.2%

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

   3. difference-of-squares 54.2%

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

   4. pow1/2 54.2%

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

   5. sqrt-pow1 54.3%

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

   6. pow2 54.3%

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

   7. metadata-eval 54.3%

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

   8. pow1/2 54.3%

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

   9. sqrt-pow1 54.1%

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

   10. pow2 54.1%

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

   11. metadata-eval 54.1%

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

4. Applied egg-rr 54.1%

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

   1. difference-of-squares 54.1%

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

   2. add-sqr-sqrt 54.0%

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

   3. flip-- 54.0%

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

   4. unpow2 54.0%

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

   5. pow-prod-up 54.5%

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

   6. metadata-eval 54.5%

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

   7. pow1/2 54.5%

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

   8. pow-prod-up 54.2%

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

   9. metadata-eval 54.2%

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

   10. pow1/2 54.2%

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

   11. add-sqr-sqrt 54.2%

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

   12. pow-prod-up 54.3%

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

6. Applied egg-rr 54.3%

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

   1. associate--r- 99.6%

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

   2. +-inverses 99.6%

      \[\leadsto \frac{\color{blue}{0} + \varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \]

   3. +-lft-identity 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in eps around 0 51.7%

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

10. Final simplification 51.7%

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

Alternative 5: 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 54.6%

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

3. Taylor expanded in x around inf 51.3%

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

4. Final simplification 51.3%

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

Alternative 6: 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 54.6%

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

   1. flip-- 54.6%

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

   2. div-inv 54.4%

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

   3. add-sqr-sqrt 54.2%

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

   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 70.9%

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

   9. hypot-define 70.9%

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

4. Applied egg-rr 70.9%

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

   1. +-inverses 70.9%

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

   2. +-commutative 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in eps around 0 0.0%

   \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \color{blue}{\left(x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]
8. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \color{blue}{\frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} \cdot 0.5}\right)} \]

   2. associate-/l* 0.0%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \color{blue}{\left(\varepsilon \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}\right)} \cdot 0.5\right)} \]

   3. associate-*r* 0.0%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \color{blue}{\varepsilon \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2}}{x} \cdot 0.5\right)}\right)} \]

   4. *-commutative 0.0%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \varepsilon \cdot \color{blue}{\left(0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}\right)}\right)} \]

   5. associate-*r/ 0.0%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \varepsilon \cdot \color{blue}{\frac{0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}\right)} \]

   6. unpow2 0.0%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \varepsilon \cdot \frac{0.5 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}{x}\right)} \]

   7. rem-square-sqrt 51.5%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \varepsilon \cdot \frac{0.5 \cdot \color{blue}{-1}}{x}\right)} \]

   8. metadata-eval 51.5%

      \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \left(x + \varepsilon \cdot \frac{\color{blue}{-0.5}}{x}\right)} \]

9. Simplified 51.5%

   \[\leadsto \left(\varepsilon + 0\right) \cdot \frac{1}{x + \color{blue}{\left(x + \varepsilon \cdot \frac{-0.5}{x}\right)}} \]

10. Taylor expanded in eps around inf 5.0%

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

    1. *-commutative 5.0%

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

12. Simplified 5.0%

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

13. Final simplification 5.0%

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

Alternative 7: 3.5% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x

Julia

function code(x, eps)
	return x
end

MATLAB

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

Wolfram

code[x_, eps_] := x

Derivation

1. Initial program 54.6%

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

   1. add-sqr-sqrt 54.2%

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

   2. add-sqr-sqrt 54.2%

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

   3. difference-of-squares 54.2%

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

   4. pow1/2 54.2%

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

   5. sqrt-pow1 54.3%

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

   6. pow2 54.3%

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

   7. metadata-eval 54.3%

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

   8. pow1/2 54.3%

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

   9. sqrt-pow1 54.1%

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

   10. pow2 54.1%

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

   11. metadata-eval 54.1%

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

4. Applied egg-rr 54.1%

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

5. Taylor expanded in x around inf 3.7%

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

6. Final simplification 3.7%

   \[\leadsto x \]
7. 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 2024067 
(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))))