ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 99.0%

Time: 8.7s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-153)
   (* eps (/ 1.0 (+ x (hypot x (sqrt (- eps))))))
   (/ eps (+ x (fma eps (/ -0.5 x) x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = eps * (1.0 / (x + hypot(x, sqrt(-eps))));
	} else {
		tmp = eps / (x + fma(eps, (-0.5 / x), x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-153)
		tmp = Float64(eps * Float64(1.0 / Float64(x + hypot(x, sqrt(Float64(-eps))))));
	else
		tmp = Float64(eps / Float64(x + fma(eps, Float64(-0.5 / 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], -1e-153], N[(eps * N[(1.0 / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.2%

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

      1. flip-- 98.1%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 97.6%

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

      4. associate--r- 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. sub-neg 99.2%

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

      8. add-sqr-sqrt 99.2%

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

      9. hypot-define 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. +-inverses 99.2%

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

      2. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in eps around 0 99.2%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 7.1%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 43.2%

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

      9. hypot-define 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-inverses 43.2%

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

      3. +-lft-identity 43.2%

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

      4. associate-*l/ 43.4%

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

      5. *-lft-identity 43.4%

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

   6. Simplified 43.4%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 99.2%

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

      5. mul-1-neg 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. distribute-lft-neg-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. associate-*r/ 99.2%

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

      10. +-commutative 99.2%

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

      11. associate-*r/ 99.2%

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

      12. associate-*l/ 99.2%

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

      13. *-commutative 99.2%

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

      14. fma-undefine 99.2%

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

   9. Simplified 99.2%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-153)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (+ x (fma eps (/ -0.5 x) x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / (x + fma(eps, (-0.5 / x), x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-153)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(x + fma(eps, Float64(-0.5 / 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], -1e-153], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.2%

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

      1. flip-- 98.1%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 97.6%

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

      4. associate--r- 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. sub-neg 99.2%

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

      8. add-sqr-sqrt 99.2%

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

      9. hypot-define 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. +-inverses 99.2%

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

      3. +-lft-identity 99.2%

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

      4. associate-*l/ 99.2%

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

      5. *-lft-identity 99.2%

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

   6. Simplified 99.2%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 7.1%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 43.2%

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

      9. hypot-define 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-inverses 43.2%

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

      3. +-lft-identity 43.2%

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

      4. associate-*l/ 43.4%

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

      5. *-lft-identity 43.4%

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

   6. Simplified 43.4%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 99.2%

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

      5. mul-1-neg 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. distribute-lft-neg-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. associate-*r/ 99.2%

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

      10. +-commutative 99.2%

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

      11. associate-*r/ 99.2%

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

      12. associate-*l/ 99.2%

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

      13. *-commutative 99.2%

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

      14. fma-undefine 99.2%

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

   9. Simplified 99.2%

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

Alternative 3: 98.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-153)
   (- x (hypot (sqrt (- eps)) x))
   (/ eps (+ x (fma eps (/ -0.5 x) x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / (x + fma(eps, (-0.5 / x), x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-153)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / Float64(x + fma(eps, Float64(-0.5 / 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], -1e-153], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.2%

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

      1. sub-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. add-sqr-sqrt 98.2%

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

      4. hypot-define 98.3%

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

   4. Applied egg-rr 98.3%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 7.1%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 43.2%

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

      9. hypot-define 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-inverses 43.2%

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

      3. +-lft-identity 43.2%

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

      4. associate-*l/ 43.4%

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

      5. *-lft-identity 43.4%

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

   6. Simplified 43.4%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 99.2%

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

      5. mul-1-neg 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. distribute-lft-neg-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. associate-*r/ 99.2%

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

      10. +-commutative 99.2%

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

      11. associate-*r/ 99.2%

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

      12. associate-*l/ 99.2%

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

      13. *-commutative 99.2%

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

      14. fma-undefine 99.2%

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

   9. Simplified 99.2%

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

Alternative 4: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{fma}\left(\varepsilon, \frac{-0.5}{x}, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-153) t_0 (/ eps (+ x (fma eps (/ -0.5 x) x))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x + fma(eps, (-0.5 / x), x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + fma(eps, Float64(-0.5 / x), x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-153], t$95$0, N[(eps / N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.2%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 7.1%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 43.2%

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

      9. hypot-define 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-inverses 43.2%

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

      3. +-lft-identity 43.2%

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

      4. associate-*l/ 43.4%

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

      5. *-lft-identity 43.4%

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

   6. Simplified 43.4%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 99.2%

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

      5. mul-1-neg 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. distribute-lft-neg-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. associate-*r/ 99.2%

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

      10. +-commutative 99.2%

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

      11. associate-*r/ 99.2%

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

      12. associate-*l/ 99.2%

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

      13. *-commutative 99.2%

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

      14. fma-undefine 99.2%

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

   9. Simplified 99.2%

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

Alternative 5: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-153) t_0 (/ eps (+ (* x 2.0) (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-153) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-153)) then
        tmp = t_0
    else
        tmp = eps / ((x * 2.0d0) + ((-0.5d0) * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-153) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-153:
		tmp = t_0
	else:
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(-0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-153)
		tmp = t_0;
	else
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-153], t$95$0, N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 98.2%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. flip-- 7.1%

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

      2. div-inv 7.1%

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

      3. add-sqr-sqrt 7.1%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 43.2%

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

      9. hypot-define 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-inverses 43.2%

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

      3. +-lft-identity 43.2%

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

      4. associate-*l/ 43.4%

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

      5. *-lft-identity 43.4%

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

   6. Simplified 43.4%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. *-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 99.2%

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

      8. mul-1-neg 99.2%

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

      9. distribute-lft-neg-in 99.2%

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

      10. distribute-frac-neg 99.2%

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

      11. associate-*l/ 99.2%

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

      12. distribute-rgt-neg-in 99.2%

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

      13. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-115}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 7.8e-115)
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* -0.5 (/ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7.8e-115) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.8d-115) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / ((x * 2.0d0) + ((-0.5d0) * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.8e-115) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 7.8e-115:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7.8e-115)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(-0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.8e-115)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7.8e-115], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.7999999999999997e-115

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 95.6%

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

      1. neg-mul-1 95.6%

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

   5. Simplified 95.6%

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

   if 7.7999999999999997e-115 < x

   1. Initial program 24.2%

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

      1. flip-- 24.3%

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

      2. div-inv 24.3%

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

      3. add-sqr-sqrt 24.2%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 55.1%

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

      9. hypot-define 55.1%

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

   4. Applied egg-rr 55.1%

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

      1. *-commutative 55.1%

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

      2. +-inverses 55.1%

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

      3. +-lft-identity 55.1%

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

      4. associate-*l/ 55.2%

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

      5. *-lft-identity 55.2%

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

   6. Simplified 55.2%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r/ 0.0%

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

      4. *-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 82.8%

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

      8. mul-1-neg 82.8%

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

      9. distribute-lft-neg-in 82.8%

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

      10. distribute-frac-neg 82.8%

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

      11. associate-*l/ 82.8%

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

      12. distribute-rgt-neg-in 82.8%

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

      13. metadata-eval 82.8%

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

   9. Simplified 82.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-115}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

   1. flip-- 57.9%

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

   2. div-inv 57.8%

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

   3. add-sqr-sqrt 57.6%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 74.5%

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

   9. hypot-define 74.5%

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

4. Applied egg-rr 74.5%

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

   1. *-commutative 74.5%

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

   2. +-inverses 74.5%

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

   3. +-lft-identity 74.5%

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

   4. associate-*l/ 74.5%

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

   5. *-lft-identity 74.5%

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

6. Simplified 74.5%

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

7. Taylor expanded in eps around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. *-commutative 0.0%

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

   3. associate-*r/ 0.0%

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

   4. *-commutative 0.0%

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

   5. *-commutative 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 49.0%

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

   8. mul-1-neg 49.0%

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

   9. distribute-lft-neg-in 49.0%

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

   10. distribute-frac-neg 49.0%

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

   11. associate-*l/ 49.0%

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

   12. distribute-rgt-neg-in 49.0%

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

   13. metadata-eval 49.0%

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

9. Simplified 49.0%

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

10. Final simplification 49.0%

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

Alternative 8: 44.7% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around inf 48.3%

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

4. Final simplification 48.3%

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

Alternative 9: 4.3% accurate, 107.0× 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 57.9%

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

3. Taylor expanded in x around inf 4.3%

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

4. Taylor expanded in x around 0 4.3%

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

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024132 
(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))))