ENA, Section 1.4, Exercise 4d

Percentage Accurate: 89.5% → 90.4%

Time: 12.0s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% 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: 90.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.7 \cdot 10^{-78}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \frac{\sqrt{{x}^{6} - {\varepsilon}^{3}}}{\sqrt{{x}^{4} + \mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon \cdot {x}^{2}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 3.7e-78)
   (- x (sqrt (- (* x x) eps)))
   (/
    eps
    (+
     x
     (/
      (sqrt (- (pow x 6.0) (pow eps 3.0)))
      (sqrt (+ (pow x 4.0) (fma eps eps (* eps (pow x 2.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 3.7e-78) {
		tmp = x - sqrt(((x * x) - eps));
	} else {
		tmp = eps / (x + (sqrt((pow(x, 6.0) - pow(eps, 3.0))) / sqrt((pow(x, 4.0) + fma(eps, eps, (eps * pow(x, 2.0)))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 3.7e-78)
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(sqrt(Float64((x ^ 6.0) - (eps ^ 3.0))) / sqrt(Float64((x ^ 4.0) + fma(eps, eps, Float64(eps * (x ^ 2.0))))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 3.7e-78], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(N[Sqrt[N[(N[Power[x, 6.0], $MachinePrecision] - N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[x, 4.0], $MachinePrecision] + N[(eps * eps + N[(eps * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.70000000000000006e-78

   1. Initial program 91.1%

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

   if 3.70000000000000006e-78 < eps

   1. Initial program 21.5%

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

      1. flip-- 22.0%

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

      2. div-inv 22.0%

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

      3. add-sqr-sqrt 24.8%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. +-lft-identity 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. /-rgt-identity 0.0%

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

   6. Simplified 0.0%

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

      1. hypot-udef 0.0%

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

      2. unpow2 0.0%

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

      3. add-sqr-sqrt 99.8%

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

      4. sub-neg 99.8%

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

      5. flip3-- 99.8%

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

      6. sqrt-div 99.6%

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

      7. pow-pow 99.8%

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

      8. metadata-eval 99.8%

         \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{\color{blue}{6}} - {\varepsilon}^{3}}}{\sqrt{{x}^{2} \cdot {x}^{2} + \left(\varepsilon \cdot \varepsilon + {x}^{2} \cdot \varepsilon\right)}}} \]

      9. pow-prod-up 99.8%

         \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{6} - {\varepsilon}^{3}}}{\sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} + \left(\varepsilon \cdot \varepsilon + {x}^{2} \cdot \varepsilon\right)}}} \]

      10. metadata-eval 99.8%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{6} - {\varepsilon}^{3}}}{\sqrt{{x}^{\color{blue}{4}} + \left(\varepsilon \cdot \varepsilon + {x}^{2} \cdot \varepsilon\right)}}} \]

      11. fma-def 99.8%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{6} - {\varepsilon}^{3}}}{\sqrt{{x}^{4} + \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon, {x}^{2} \cdot \varepsilon\right)}}}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\frac{\sqrt{{x}^{6} - {\varepsilon}^{3}}}{\sqrt{{x}^{4} + \mathsf{fma}\left(\varepsilon, \varepsilon, {x}^{2} \cdot \varepsilon\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.7 \cdot 10^{-78}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \frac{\sqrt{{x}^{6} - {\varepsilon}^{3}}}{\sqrt{{x}^{4} + \mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon \cdot {x}^{2}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-77}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.4e-77)
   (- x (sqrt (- (* x x) eps)))
   (/
    eps
    (+ x (/ (sqrt (- (pow x 4.0) (pow eps 2.0))) (hypot x (sqrt eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.4e-77) {
		tmp = x - sqrt(((x * x) - eps));
	} else {
		tmp = eps / (x + (sqrt((pow(x, 4.0) - pow(eps, 2.0))) / hypot(x, sqrt(eps))));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.4e-77) {
		tmp = x - Math.sqrt(((x * x) - eps));
	} else {
		tmp = eps / (x + (Math.sqrt((Math.pow(x, 4.0) - Math.pow(eps, 2.0))) / Math.hypot(x, Math.sqrt(eps))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1.4e-77:
		tmp = x - math.sqrt(((x * x) - eps))
	else:
		tmp = eps / (x + (math.sqrt((math.pow(x, 4.0) - math.pow(eps, 2.0))) / math.hypot(x, math.sqrt(eps))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.4e-77)
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(sqrt(Float64((x ^ 4.0) - (eps ^ 2.0))) / hypot(x, sqrt(eps)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.4e-77)
		tmp = x - sqrt(((x * x) - eps));
	else
		tmp = eps / (x + (sqrt(((x ^ 4.0) - (eps ^ 2.0))) / hypot(x, sqrt(eps))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.4e-77], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(N[Sqrt[N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[x ^ 2 + N[Sqrt[eps], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.4e-77

   1. Initial program 91.1%

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

   if 1.4e-77 < eps

   1. Initial program 21.5%

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

      1. flip-- 22.0%

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

      2. div-inv 22.0%

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

      3. add-sqr-sqrt 24.8%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. +-lft-identity 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. /-rgt-identity 0.0%

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

   6. Simplified 0.0%

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

      1. hypot-udef 0.0%

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

      2. unpow2 0.0%

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

      3. add-sqr-sqrt 99.8%

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

      4. sub-neg 99.8%

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

      5. flip-- 99.8%

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

      6. sqrt-div 99.8%

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

      7. pow-prod-up 99.6%

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

      8. metadata-eval 99.6%

         \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{\color{blue}{4}} - \varepsilon \cdot \varepsilon}}{\sqrt{{x}^{2} + \varepsilon}}} \]

      9. pow2 99.6%

         \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - \color{blue}{{\varepsilon}^{2}}}}{\sqrt{{x}^{2} + \varepsilon}}} \]

      10. unpow2 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\sqrt{\color{blue}{x \cdot x} + \varepsilon}}} \]

      11. add-sqr-sqrt 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\sqrt{x \cdot x + \color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}}}} \]

      12. sqrt-unprod 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\sqrt{x \cdot x + \color{blue}{\sqrt{\varepsilon \cdot \varepsilon}}}}} \]

      13. sqr-neg 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\sqrt{x \cdot x + \sqrt{\color{blue}{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}}}}} \]

      14. sqrt-unprod 0.0%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\sqrt{x \cdot x + \color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}}} \]

      15. hypot-udef 0.0%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}}} \]

      16. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}\right)}} \]

      17. sqrt-unprod 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{\sqrt{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}}}\right)}} \]

      18. sqr-neg 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}}\right)}} \]

      19. sqrt-unprod 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}}\right)}} \]

      20. add-sqr-sqrt 99.6%

          \[\leadsto \frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{\varepsilon}}\right)}} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-77}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \frac{\sqrt{{x}^{4} - {\varepsilon}^{2}}}{\mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.4 \cdot 10^{-78}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\left(\frac{\varepsilon}{x}\right)}^{2}}{x}, \frac{\varepsilon}{x} \cdot \left(-0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 5.4e-78)
   (- x (sqrt (- (* x x) eps)))
   (/
    eps
    (+ x (+ x (fma -0.125 (/ (pow (/ eps x) 2.0) x) (* (/ eps x) (- 0.5))))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 5.4e-78) {
		tmp = x - sqrt(((x * x) - eps));
	} else {
		tmp = eps / (x + (x + fma(-0.125, (pow((eps / x), 2.0) / x), ((eps / x) * -0.5))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 5.4e-78)
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + fma(-0.125, Float64((Float64(eps / x) ^ 2.0) / x), Float64(Float64(eps / x) * Float64(-0.5))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 5.4e-78], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(-0.125 * N[(N[Power[N[(eps / x), $MachinePrecision], 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.39999999999999987e-78

   1. Initial program 91.1%

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

   if 5.39999999999999987e-78 < eps

   1. Initial program 21.5%

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

      1. flip-- 22.0%

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

      2. div-inv 22.0%

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

      3. add-sqr-sqrt 24.8%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. +-lft-identity 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. /-rgt-identity 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \left(-0.125 \cdot \frac{{\varepsilon}^{2} \cdot {\left(\sqrt{-1}\right)}^{4}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)\right)}} \]
   8. Step-by-step derivation

      1. fma-def 0.0%

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

      2. unpow2 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

      11. swap-sqr 0.0%

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

      12. *-rgt-identity 0.0%

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

      13. *-rgt-identity 0.0%

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

      14. unpow2 0.0%

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

      15. *-commutative 0.0%

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

      16. unpow2 0.0%

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

      17. rem-square-sqrt 98.4%

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

      18. mul-1-neg 98.4%

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

   9. Simplified 98.4%

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

       1. *-un-lft-identity 98.4%

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

       2. cube-mult 98.4%

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

       3. unpow2 98.4%

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

       4. times-frac 98.4%

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

       5. unpow2 98.4%

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

       6. unpow2 98.4%

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

       7. frac-times 98.4%

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

       8. pow2 98.4%

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

   11. Applied egg-rr 98.4%

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

       1. associate-*l/ 98.4%

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

       2. *-lft-identity 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.4 \cdot 10^{-78}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \mathsf{fma}\left(-0.125, \frac{{\left(\frac{\varepsilon}{x}\right)}^{2}}{x}, \frac{\varepsilon}{x} \cdot \left(-0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-114} \lor \neg \left(x \leq 3.7 \cdot 10^{-91}\right) \land x \leq 9.8 \cdot 10^{-66}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x 1.9e-114) (and (not (<= x 3.7e-91)) (<= x 9.8e-66)))
   (- x (sqrt (- eps)))
   (- x (+ x (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= 1.9e-114) || (!(x <= 3.7e-91) && (x <= 9.8e-66))) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = x - (x + ((eps / x) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= 1.9d-114) .or. (.not. (x <= 3.7d-91)) .and. (x <= 9.8d-66)) then
        tmp = x - sqrt(-eps)
    else
        tmp = x - (x + ((eps / x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= 1.9e-114) || (!(x <= 3.7e-91) && (x <= 9.8e-66))) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = x - (x + ((eps / x) * -0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= 1.9e-114) or (not (x <= 3.7e-91) and (x <= 9.8e-66)):
		tmp = x - math.sqrt(-eps)
	else:
		tmp = x - (x + ((eps / x) * -0.5))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= 1.9e-114) || (!(x <= 3.7e-91) && (x <= 9.8e-66)))
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(x - Float64(x + Float64(Float64(eps / x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 1.9e-114) || (~((x <= 3.7e-91)) && (x <= 9.8e-66)))
		tmp = x - sqrt(-eps);
	else
		tmp = x - (x + ((eps / x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, 1.9e-114], And[N[Not[LessEqual[x, 3.7e-91]], $MachinePrecision], LessEqual[x, 9.8e-66]]], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(x - N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999e-114 or 3.7000000000000002e-91 < x < 9.8000000000000001e-66

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0 86.8%

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

      1. neg-mul-1 86.8%

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

   5. Simplified 86.8%

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

   if 1.8999999999999999e-114 < x < 3.7000000000000002e-91 or 9.8000000000000001e-66 < x

   1. Initial program 82.1%

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

   3. Taylor expanded in x around inf 69.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-114} \lor \neg \left(x \leq 3.7 \cdot 10^{-91}\right) \land x \leq 9.8 \cdot 10^{-66}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.15e-77)
   (- x (sqrt (- (* x x) eps)))
   (/ eps (+ (* (/ eps x) -0.5) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.15e-77) {
		tmp = x - sqrt(((x * x) - eps));
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.15d-77) then
        tmp = x - sqrt(((x * x) - eps))
    else
        tmp = eps / (((eps / x) * (-0.5d0)) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.15e-77) {
		tmp = x - Math.sqrt(((x * x) - eps));
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1.15e-77:
		tmp = x - math.sqrt(((x * x) - eps))
	else:
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.15e-77)
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	else
		tmp = Float64(eps / Float64(Float64(Float64(eps / x) * -0.5) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.15e-77)
		tmp = x - sqrt(((x * x) - eps));
	else
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.15e-77], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.14999999999999999e-77

   1. Initial program 91.1%

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

   if 1.14999999999999999e-77 < eps

   1. Initial program 21.5%

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

      1. flip-- 22.0%

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

      2. div-inv 22.0%

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

      3. add-sqr-sqrt 24.8%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 0.0%

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

      9. hypot-def 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. +-lft-identity 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. /-rgt-identity 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 95.7%

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

      4. mul-1-neg 95.7%

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

   9. Simplified 95.7%

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

   10. Taylor expanded in x around 0 95.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\varepsilon}{x} \cdot -0.5\\ \mathbf{if}\;\varepsilon \leq -6 \cdot 10^{-96} \lor \neg \left(\varepsilon \leq 4.6 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\varepsilon}{t\_0 + x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (/ eps x) -0.5)))
   (if (or (<= eps -6e-96) (not (<= eps 4.6e-78)))
     (/ eps (+ t_0 (* x 2.0)))
     (- x (+ x t_0)))))

C

double code(double x, double eps) {
	double t_0 = (eps / x) * -0.5;
	double tmp;
	if ((eps <= -6e-96) || !(eps <= 4.6e-78)) {
		tmp = eps / (t_0 + (x * 2.0));
	} else {
		tmp = x - (x + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (eps / x) * (-0.5d0)
    if ((eps <= (-6d-96)) .or. (.not. (eps <= 4.6d-78))) then
        tmp = eps / (t_0 + (x * 2.0d0))
    else
        tmp = x - (x + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (eps / x) * -0.5;
	double tmp;
	if ((eps <= -6e-96) || !(eps <= 4.6e-78)) {
		tmp = eps / (t_0 + (x * 2.0));
	} else {
		tmp = x - (x + t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (eps / x) * -0.5
	tmp = 0
	if (eps <= -6e-96) or not (eps <= 4.6e-78):
		tmp = eps / (t_0 + (x * 2.0))
	else:
		tmp = x - (x + t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps / x) * -0.5)
	tmp = 0.0
	if ((eps <= -6e-96) || !(eps <= 4.6e-78))
		tmp = Float64(eps / Float64(t_0 + Float64(x * 2.0)));
	else
		tmp = Float64(x - Float64(x + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps / x) * -0.5;
	tmp = 0.0;
	if ((eps <= -6e-96) || ~((eps <= 4.6e-78)))
		tmp = eps / (t_0 + (x * 2.0));
	else
		tmp = x - (x + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]}, If[Or[LessEqual[eps, -6e-96], N[Not[LessEqual[eps, 4.6e-78]], $MachinePrecision]], N[(eps / N[(t$95$0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -6e-96 or 4.6000000000000004e-78 < eps

   1. Initial program 84.0%

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

      1. flip-- 84.0%

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

      2. div-inv 83.7%

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

      3. add-sqr-sqrt 83.8%

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

      4. associate--r- 96.8%

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

      5. pow2 96.8%

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

      6. pow2 96.8%

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

      7. sub-neg 96.8%

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

      8. add-sqr-sqrt 86.6%

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

      9. hypot-def 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. +-inverses 86.6%

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

      2. +-lft-identity 86.6%

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

      3. associate-*r/ 86.6%

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

      4. associate-/l* 86.6%

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

      5. /-rgt-identity 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 23.8%

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

      4. mul-1-neg 23.8%

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

   9. Simplified 23.8%

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

   10. Taylor expanded in x around 0 23.8%

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

   if -6e-96 < eps < 4.6000000000000004e-78

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 48.6%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6 \cdot 10^{-96} \lor \neg \left(\varepsilon \leq 4.6 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-96} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.1e-96) (not (<= eps 6.2e-78))) (* (/ eps x) 0.5) (- x x)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-96) || !(eps <= 6.2e-78)) {
		tmp = (eps / x) * 0.5;
	} else {
		tmp = x - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.1d-96)) .or. (.not. (eps <= 6.2d-78))) then
        tmp = (eps / x) * 0.5d0
    else
        tmp = x - x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-96) || !(eps <= 6.2e-78)) {
		tmp = (eps / x) * 0.5;
	} else {
		tmp = x - x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.1e-96) or not (eps <= 6.2e-78):
		tmp = (eps / x) * 0.5
	else:
		tmp = x - x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.1e-96) || !(eps <= 6.2e-78))
		tmp = Float64(Float64(eps / x) * 0.5);
	else
		tmp = Float64(x - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.1e-96) || ~((eps <= 6.2e-78)))
		tmp = (eps / x) * 0.5;
	else
		tmp = x - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.1e-96], N[Not[LessEqual[eps, 6.2e-78]], $MachinePrecision]], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision], N[(x - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.0999999999999999e-96 or 6.20000000000000035e-78 < eps

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf 22.6%

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

   if -1.0999999999999999e-96 < eps < 6.20000000000000035e-78

   1. Initial program 91.0%

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

   3. Taylor expanded in x around inf 45.8%

      \[\leadsto x - \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-96} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.45 \cdot 10^{-77}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.45e-77) (- x (+ x (* (/ eps x) -0.5))) (* (/ eps x) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.45e-77) {
		tmp = x - (x + ((eps / x) * -0.5));
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.45d-77) then
        tmp = x - (x + ((eps / x) * (-0.5d0)))
    else
        tmp = (eps / x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.45e-77) {
		tmp = x - (x + ((eps / x) * -0.5));
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1.45e-77:
		tmp = x - (x + ((eps / x) * -0.5))
	else:
		tmp = (eps / x) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.45e-77)
		tmp = Float64(x - Float64(x + Float64(Float64(eps / x) * -0.5)));
	else
		tmp = Float64(Float64(eps / x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.45e-77)
		tmp = x - (x + ((eps / x) * -0.5));
	else
		tmp = (eps / x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.45e-77], N[(x - N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.4499999999999999e-77

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 37.5%

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

   if 1.4499999999999999e-77 < eps

   1. Initial program 21.5%

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

   3. Taylor expanded in x around inf 89.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.45 \cdot 10^{-77}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 5.2% 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 88.9%

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

   1. flip-- 88.9%

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

   2. div-inv 88.7%

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

   3. add-sqr-sqrt 88.8%

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

   4. associate--r- 70.5%

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

   5. pow2 70.4%

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

   6. pow2 70.5%

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

   7. sub-neg 70.5%

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

   8. add-sqr-sqrt 61.7%

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

   9. hypot-def 61.7%

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

4. Applied egg-rr 61.7%

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

   1. +-inverses 61.7%

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

   2. +-lft-identity 61.7%

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

   3. associate-*r/ 61.7%

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

   4. associate-/l* 61.7%

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

   5. /-rgt-identity 61.7%

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

6. Simplified 61.7%

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

7. Taylor expanded in x around inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 19.0%

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

   4. mul-1-neg 19.0%

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

9. Simplified 19.0%

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

10. Taylor expanded in eps around inf 5.5%

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

    1. *-commutative 5.5%

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

12. Simplified 5.5%

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

13. Final simplification 5.5%

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

Alternative 10: 31.7% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in x around inf 33.4%

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

4. Final simplification 33.4%

   \[\leadsto x - x \]
5. Add Preprocessing

Alternative 11: 3.4% 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 88.9%

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

3. Taylor expanded in x around 0 52.9%

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

   1. neg-mul-1 52.9%

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

5. Simplified 52.9%

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

6. Taylor expanded in x around inf 3.4%

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

7. Final simplification 3.4%

   \[\leadsto x \]
8. Add Preprocessing

Developer target: 72.1% 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 2024031 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4d"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))

  :herbie-target
  (/ eps (+ x (sqrt (- (* x x) eps))))

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