ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 98.3%

Time: 10.8s

Alternatives: 12

Speedup: 0.5×

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: 98.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (* eps (fma x (/ x eps) -1.0))))
   (/ (fma eps 0.5 (/ (* 0.125 (* eps eps)) (* x x))) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt((eps * fma(x, (x / eps), -1.0)));
	} else {
		tmp = fma(eps, 0.5, ((0.125 * (eps * eps)) / (x * x))) / x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(Float64(eps * fma(x, Float64(x / eps), -1.0))));
	else
		tmp = Float64(fma(eps, 0.5, Float64(Float64(0.125 * Float64(eps * eps)) / Float64(x * x))) / x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[(eps * N[(x * N[(x / eps), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps * 0.5 + N[(N[(0.125 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right) + \frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}}}{x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}}{x} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)}}{x} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)}}{x} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)}{x}} \]

   5. Simplified 99.9%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (* eps (fma x (/ x eps) -1.0))))
   (/ (* eps (fma eps (/ 0.125 (* x x)) 0.5)) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt((eps * fma(x, (x / eps), -1.0)));
	} else {
		tmp = (eps * fma(eps, (0.125 / (x * x)), 0.5)) / x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(Float64(eps * fma(x, Float64(x / eps), -1.0))));
	else
		tmp = Float64(Float64(eps * fma(eps, Float64(0.125 / Float64(x * x)), 0.5)) / x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[(eps * N[(x * N[(x / eps), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps * N[(eps * N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. neg-lowering-neg.f64 6.5

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

   4. Applied egg-rr 6.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)} - \frac{-1}{2} \cdot \varepsilon}{x} \]

      3. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right)}}{x} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} + \frac{-1}{2} \cdot \varepsilon\right)\right)}}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)}\right)}{x} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)}{x}} \]

   7. Simplified 99.9%

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

Alternative 3: 98.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (* eps (fma x (/ x eps) -1.0))))
   (* (fma eps (/ 0.125 (* x x)) 0.5) (/ eps x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt((eps * fma(x, (x / eps), -1.0)));
	} else {
		tmp = fma(eps, (0.125 / (x * x)), 0.5) * (eps / x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(Float64(eps * fma(x, Float64(x / eps), -1.0))));
	else
		tmp = Float64(fma(eps, Float64(0.125 / Float64(x * x)), 0.5) * Float64(eps / x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[(eps * N[(x * N[(x / eps), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps * N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. neg-lowering-neg.f64 6.5

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

   4. Applied egg-rr 6.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)} - \frac{-1}{2} \cdot \varepsilon}{x} \]

      3. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right)}}{x} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} + \frac{-1}{2} \cdot \varepsilon\right)\right)}}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)}\right)}{x} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)}{x}} \]

   7. Simplified 99.9%

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \frac{\frac{1}{8}}{x \cdot x} + \frac{1}{2}\right) \cdot \varepsilon}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \frac{\frac{1}{8}}{x \cdot x} + \frac{1}{2}\right) \cdot \frac{\varepsilon}{x}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \frac{\frac{1}{8}}{x \cdot x} + \frac{1}{2}\right) \cdot \frac{\varepsilon}{x}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{\frac{1}{8}}{x \cdot x}, \frac{1}{2}\right)} \cdot \frac{\varepsilon}{x} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{\frac{1}{8}}{x \cdot x}}, \frac{1}{2}\right) \cdot \frac{\varepsilon}{x} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{\frac{1}{8}}{\color{blue}{x \cdot x}}, \frac{1}{2}\right) \cdot \frac{\varepsilon}{x} \]

      7. /-lowering-/.f64 99.9

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

   9. Applied egg-rr 99.9%

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

Alternative 4: 98.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (* eps (fma x (/ x eps) -1.0))))
   (* eps (/ (fma eps (/ 0.125 (* x x)) 0.5) x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt((eps * fma(x, (x / eps), -1.0)));
	} else {
		tmp = eps * (fma(eps, (0.125 / (x * x)), 0.5) / x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(Float64(eps * fma(x, Float64(x / eps), -1.0))));
	else
		tmp = Float64(eps * Float64(fma(eps, Float64(0.125 / Float64(x * x)), 0.5) / x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[(eps * N[(x * N[(x / eps), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(eps * N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. neg-lowering-neg.f64 6.5

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

   4. Applied egg-rr 6.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8}\right)\right)} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)} - \frac{-1}{2} \cdot \varepsilon}{x} \]

      3. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right)}}{x} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} + \frac{-1}{2} \cdot \varepsilon\right)\right)}}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)}\right)}{x} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}}\right)\right)}{x}} \]

   7. Simplified 99.9%

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \frac{\varepsilon \cdot \frac{\frac{1}{8}}{x \cdot x} + \frac{1}{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\varepsilon \cdot \frac{\frac{1}{8}}{x \cdot x} + \frac{1}{2}}{x} \cdot \varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\varepsilon \cdot \frac{\frac{1}{8}}{x \cdot x} + \frac{1}{2}}{x} \cdot \varepsilon} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\varepsilon \cdot \frac{\frac{1}{8}}{x \cdot x} + \frac{1}{2}}{x}} \cdot \varepsilon \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{\frac{1}{8}}{x \cdot x}, \frac{1}{2}\right)}}{x} \cdot \varepsilon \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\varepsilon, \color{blue}{\frac{\frac{1}{8}}{x \cdot x}}, \frac{1}{2}\right)}{x} \cdot \varepsilon \]

      7. *-lowering-*.f64 99.5

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 98.9%

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

Alternative 5: 98.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (* eps (fma x (/ x eps) -1.0))))
   (/ (* eps 0.5) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt((eps * fma(x, (x / eps), -1.0)));
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(Float64(eps * fma(x, Float64(x / eps), -1.0))));
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[(eps * N[(x * N[(x / eps), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \varepsilon}{x}} \]

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\varepsilon \cdot \color{blue}{\frac{1}{2}}}{x} \]

      8. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 6: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-153) t_0 (/ (* eps 0.5) x))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-153) {
		tmp = t_0;
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-2d-153)) then
        tmp = t_0
    else
        tmp = (eps * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-153) {
		tmp = t_0;
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-153:
		tmp = t_0
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -2e-153)
		tmp = t_0;
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -2e-153)
		tmp = t_0;
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-153], t$95$0, N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \varepsilon}{x}} \]

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\varepsilon \cdot \color{blue}{\frac{1}{2}}}{x} \]

      8. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 7: 96.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (fabs eps)))
   (/ (* eps 0.5) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt(fabs(eps));
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-2d-153)) then
        tmp = x - sqrt(abs(eps))
    else
        tmp = (eps * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - Math.sqrt(Math.abs(eps));
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-153:
		tmp = x - math.sqrt(math.fabs(eps))
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(abs(eps)));
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153)
		tmp = x - sqrt(abs(eps));
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[Abs[eps], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 95.8

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

   5. Simplified 95.8%

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

      1. flip-- N/A

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

      2. rem-square-sqrt N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 95.1

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

   7. Applied egg-rr 95.1%

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

      1. rem-square-sqrt N/A

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

      2. flip-- N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqr-neg N/A

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

      8. rem-sqrt-square N/A

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

      9. fabs-lowering-fabs.f64 95.8

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

   9. Applied egg-rr 95.8%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \varepsilon}{x}} \]

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\varepsilon \cdot \color{blue}{\frac{1}{2}}}{x} \]

      8. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

Alternative 8: 96.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (fabs eps)))
   (* eps (/ 0.5 x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt(fabs(eps));
	} else {
		tmp = eps * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-2d-153)) then
        tmp = x - sqrt(abs(eps))
    else
        tmp = eps * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - Math.sqrt(Math.abs(eps));
	} else {
		tmp = eps * (0.5 / x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-153:
		tmp = x - math.sqrt(math.fabs(eps))
	else:
		tmp = eps * (0.5 / x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(abs(eps)));
	else
		tmp = Float64(eps * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153)
		tmp = x - sqrt(abs(eps));
	else
		tmp = eps * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[Abs[eps], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 95.8

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

   5. Simplified 95.8%

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

      1. flip-- N/A

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

      2. rem-square-sqrt N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 95.1

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

   7. Applied egg-rr 95.1%

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

      1. rem-square-sqrt N/A

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

      2. flip-- N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqr-neg N/A

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

      8. rem-sqrt-square N/A

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

      9. fabs-lowering-fabs.f64 95.8

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

   9. Applied egg-rr 95.8%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \varepsilon}{x}} \]

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\varepsilon \cdot \color{blue}{\frac{1}{2}}}{x} \]

      8. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\varepsilon \cdot \frac{\frac{1}{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x} \cdot \varepsilon} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x} \cdot \varepsilon} \]

      4. /-lowering-/.f64 98.9

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\left|\varepsilon\right|}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\left|\varepsilon\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (fabs eps)))
   (/ eps x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt(fabs(eps));
	} else {
		tmp = 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 - sqrt(((x * x) - eps))) <= (-2d-153)) then
        tmp = x - sqrt(abs(eps))
    else
        tmp = eps / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - Math.sqrt(Math.abs(eps));
	} else {
		tmp = eps / x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-153:
		tmp = x - math.sqrt(math.fabs(eps))
	else:
		tmp = eps / x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(abs(eps)));
	else
		tmp = Float64(eps / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153)
		tmp = x - sqrt(abs(eps));
	else
		tmp = eps / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[Abs[eps], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 95.8

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

   5. Simplified 95.8%

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

      1. flip-- N/A

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

      2. rem-square-sqrt N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 95.1

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

   7. Applied egg-rr 95.1%

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

      1. rem-square-sqrt N/A

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

      2. flip-- N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqr-neg N/A

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

      8. rem-sqrt-square N/A

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

      9. fabs-lowering-fabs.f64 95.8

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

   9. Applied egg-rr 95.8%

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 1.1

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

   5. Simplified 1.1%

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

      1. flip-- N/A

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

      2. rem-square-sqrt N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 1.1

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

   7. Applied egg-rr 1.1%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 8.6%

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

   11. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 18.9

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

   13. Simplified 18.9%

       \[\leadsto \color{blue}{\frac{\varepsilon}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;-\sqrt{\left|\varepsilon\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- (sqrt (fabs eps)))
   (/ eps x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = -sqrt(fabs(eps));
	} else {
		tmp = 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 - sqrt(((x * x) - eps))) <= (-2d-153)) then
        tmp = -sqrt(abs(eps))
    else
        tmp = eps / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = -Math.sqrt(Math.abs(eps));
	} else {
		tmp = eps / x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-153:
		tmp = -math.sqrt(math.fabs(eps))
	else:
		tmp = eps / x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(-sqrt(abs(eps)));
	else
		tmp = Float64(eps / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153)
		tmp = -sqrt(abs(eps));
	else
		tmp = eps / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], (-N[Sqrt[N[Abs[eps], $MachinePrecision]], $MachinePrecision]), N[(eps / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 95.8

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

   5. Simplified 95.8%

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

      1. flip-- N/A

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

      2. rem-square-sqrt N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 95.1

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

   7. Applied egg-rr 95.1%

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

      1. rem-square-sqrt N/A

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

      2. flip-- N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqr-neg N/A

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

      8. rem-sqrt-square N/A

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

      9. fabs-lowering-fabs.f64 95.8

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

   9. Applied egg-rr 95.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{-1 \cdot \sqrt{\left|\varepsilon\right|}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\left|\varepsilon\right|}\right)} \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\left|\varepsilon\right|}\right)} \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\left|\varepsilon\right|}}\right) \]

       4. fabs-lowering-fabs.f64 94.8

          \[\leadsto -\sqrt{\color{blue}{\left|\varepsilon\right|}} \]

   12. Simplified 94.8%

       \[\leadsto \color{blue}{-\sqrt{\left|\varepsilon\right|}} \]

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 1.1

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

   5. Simplified 1.1%

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

      1. flip-- N/A

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

      2. rem-square-sqrt N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. neg-lowering-neg.f64 1.1

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

   7. Applied egg-rr 1.1%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 8.6%

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

   11. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 18.9

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

   13. Simplified 18.9%

       \[\leadsto \color{blue}{\frac{\varepsilon}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 11.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 58.8

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

5. Simplified 58.8%

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

   1. flip-- N/A

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

   2. rem-square-sqrt N/A

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

   3. /-lowering-/.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. neg-lowering-neg.f64 58.4

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

7. Applied egg-rr 58.4%

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

8. Taylor expanded in x around 0

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

10. Simplified 61.6%

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

11. Taylor expanded in x around inf

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

    1. /-lowering-/.f64 11.3

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

13. Simplified 11.3%

    \[\leadsto \color{blue}{\frac{\varepsilon}{x}} \]
14. Add Preprocessing

Alternative 12: 4.3% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 4.4%

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

   1. +-inverses 4.4

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

7. Applied egg-rr 4.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.7× 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 2024199 
(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
  (! :herbie-platform default (/ eps (+ x (sqrt (- (* x x) eps)))))

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