ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.5% → 98.5%

Time: 7.2s

Alternatives: 8

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: 61.5% 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: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{2}} \cdot \frac{\varepsilon}{x}, \mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-152)
   (- x (hypot (sqrt (- eps)) x))
   (/
    eps
    (fma
     -0.125
     (* (/ eps (pow x 2.0)) (/ eps x))
     (fma x 2.0 (* eps (/ -0.5 x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-152) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / fma(-0.125, ((eps / pow(x, 2.0)) * (eps / x)), fma(x, 2.0, (eps * (-0.5 / x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-152)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / fma(-0.125, Float64(Float64(eps / (x ^ 2.0)) * Float64(eps / x)), fma(x, 2.0, Float64(eps * Float64(-0.5 / x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-152], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(-0.125 * N[(N[(eps / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0 + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000007e-152

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. add-sqr-sqrt 99.2%

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

      4. hypot-def 99.2%

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

   3. Applied egg-rr 99.2%

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

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.6%

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

      1. flip-- 8.6%

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

      2. div-inv 8.6%

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

      3. add-sqr-sqrt 8.7%

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

      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 47.4%

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

      9. hypot-def 47.4%

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

   3. Applied egg-rr 47.4%

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

      1. +-inverses 47.4%

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

      2. +-lft-identity 47.4%

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

      3. associate-*r/ 47.5%

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

      4. associate-/l* 47.5%

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

      5. /-rgt-identity 47.5%

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. +-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. fma-def 0.0%

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

      14. associate-*r/ 0.0%

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

   8. Simplified 87.5%

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

      1. unpow2 87.5%

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

      2. unpow3 87.5%

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

      3. times-frac 99.8%

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

      4. pow2 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(-0.125, \frac{\varepsilon}{{x}^{2}} \cdot \frac{\varepsilon}{x}, \mathsf{fma}\left(x, 2, \varepsilon \cdot \frac{-0.5}{x}\right)\right)}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-152)
   (- x (hypot (sqrt (- eps)) x))
   (/ eps (+ (* (/ eps x) -0.5) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-152) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-152) {
		tmp = x - Math.hypot(Math.sqrt(-eps), x);
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-152:
		tmp = x - math.hypot(math.sqrt(-eps), x)
	else:
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-152)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	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 ((x - sqrt(((x * x) - eps))) <= -1e-152)
		tmp = x - hypot(sqrt(-eps), x);
	else
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-152], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $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 (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000007e-152

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. add-sqr-sqrt 99.2%

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

      4. hypot-def 99.2%

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

   3. Applied egg-rr 99.2%

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

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.6%

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

      1. flip-- 8.6%

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

      2. div-inv 8.6%

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

      3. add-sqr-sqrt 8.7%

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

      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 47.4%

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

      9. hypot-def 47.4%

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

   3. Applied egg-rr 47.4%

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

      1. +-inverses 47.4%

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

      2. +-lft-identity 47.4%

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

      3. associate-*r/ 47.5%

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

      4. associate-/l* 47.5%

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

      5. /-rgt-identity 47.5%

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. +-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. fma-def 0.0%

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

      14. associate-*r/ 0.0%

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

   8. Simplified 87.5%

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

   9. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-152) t_0 (/ eps (+ (* (/ eps x) -0.5) (* x 2.0))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-152)) then
        tmp = t_0
    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 t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-152:
		tmp = t_0
	else:
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-152)
		tmp = t_0;
	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)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-152)
		tmp = t_0;
	else
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-152], t$95$0, 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 (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000007e-152

   1. Initial program 99.2%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.6%

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

      1. flip-- 8.6%

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

      2. div-inv 8.6%

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

      3. add-sqr-sqrt 8.7%

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

      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 47.4%

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

      9. hypot-def 47.4%

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

   3. Applied egg-rr 47.4%

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

      1. +-inverses 47.4%

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

      2. +-lft-identity 47.4%

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

      3. associate-*r/ 47.5%

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

      4. associate-/l* 47.5%

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

      5. /-rgt-identity 47.5%

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. +-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. fma-def 0.0%

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

      14. associate-*r/ 0.0%

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

   8. Simplified 87.5%

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

   9. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 4: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-122}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 9.2e-122)
   (- x (sqrt (- eps)))
   (/ eps (+ (* (/ eps x) -0.5) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 9.2e-122) {
		tmp = x - sqrt(-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 (x <= 9.2d-122) then
        tmp = x - sqrt(-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 (x <= 9.2e-122) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 9.2e-122:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 9.2e-122)
		tmp = Float64(x - sqrt(Float64(-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 (x <= 9.2e-122)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (((eps / x) * -0.5) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 9.2e-122], N[(x - N[Sqrt[(-eps)], $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 x < 9.20000000000000028e-122

   1. Initial program 96.5%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around 0 93.4%

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

      1. neg-mul-1 93.4%

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

   4. Simplified 93.4%

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

   if 9.20000000000000028e-122 < x

   1. Initial program 23.3%

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

      1. flip-- 23.2%

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

      2. div-inv 23.2%

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

      3. add-sqr-sqrt 23.2%

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

      4. associate--r- 99.6%

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

      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 58.7%

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

      9. hypot-def 58.8%

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

   3. Applied egg-rr 58.8%

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

      1. +-inverses 58.8%

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

      2. +-lft-identity 58.8%

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

      3. associate-*r/ 58.9%

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

      4. associate-/l* 58.9%

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

      5. /-rgt-identity 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. metadata-eval 0.0%

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

      4. pow-sqr 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. metadata-eval 0.0%

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

      10. *-lft-identity 0.0%

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

      11. +-commutative 0.0%

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

      12. *-commutative 0.0%

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

      13. fma-def 0.0%

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

      14. associate-*r/ 0.0%

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

   8. Simplified 78.4%

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

   9. Taylor expanded in eps around 0 84.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-122}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 5: 45.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 \cdot \frac{x}{\varepsilon} - \frac{0.5}{x}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ 1.0 (- (* 2.0 (/ x eps)) (/ 0.5 x))))

C

double code(double x, double eps) {
	return 1.0 / ((2.0 * (x / eps)) - (0.5 / x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 / ((2.0d0 * (x / eps)) - (0.5d0 / x))
end function

Java

public static double code(double x, double eps) {
	return 1.0 / ((2.0 * (x / eps)) - (0.5 / x));
}

Python

def code(x, eps):
	return 1.0 / ((2.0 * (x / eps)) - (0.5 / x))

Julia

function code(x, eps)
	return Float64(1.0 / Float64(Float64(2.0 * Float64(x / eps)) - Float64(0.5 / x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 / ((2.0 * (x / eps)) - (0.5 / x));
end

Wolfram

code[x_, eps_] := N[(1.0 / N[(N[(2.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

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

   1. flip-- 56.7%

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

   2. div-inv 56.5%

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

   3. add-sqr-sqrt 56.3%

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

   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 74.9%

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

   9. hypot-def 74.9%

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

3. Applied egg-rr 74.9%

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

   1. *-commutative 74.9%

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

   2. associate-/r/ 74.8%

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

   3. +-inverses 74.8%

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

   4. +-commutative 74.8%

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

5. Simplified 74.8%

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

6. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. associate-*r/ 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 50.7%

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

   6. metadata-eval 50.7%

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

8. Simplified 50.7%

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

9. Taylor expanded in x around 0 50.7%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{x}}} \]

11. Simplified 50.7%

    \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} - \frac{0.5}{x}}} \]

12. Final simplification 50.7%

    \[\leadsto \frac{1}{2 \cdot \frac{x}{\varepsilon} - \frac{0.5}{x}} \]

Alternative 6: 45.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

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

   1. flip-- 56.7%

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

   2. div-inv 56.5%

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

   3. add-sqr-sqrt 56.3%

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

   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 74.9%

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

   9. hypot-def 74.9%

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

3. Applied egg-rr 74.9%

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

   1. +-inverses 74.9%

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

   2. +-lft-identity 74.9%

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

   3. associate-*r/ 75.0%

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

   4. associate-/l* 75.0%

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

   5. /-rgt-identity 75.0%

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

5. Simplified 75.0%

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

6. Taylor expanded in x around inf 0.0%

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

   1. fma-def 0.0%

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

   2. *-commutative 0.0%

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

   3. metadata-eval 0.0%

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

   4. pow-sqr 0.0%

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

   5. unpow2 0.0%

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

   6. rem-square-sqrt 0.0%

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

   7. unpow2 0.0%

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

   8. rem-square-sqrt 0.0%

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

   9. metadata-eval 0.0%

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

   10. *-lft-identity 0.0%

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

   11. +-commutative 0.0%

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

   12. *-commutative 0.0%

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

   13. fma-def 0.0%

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

   14. associate-*r/ 0.0%

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

8. Simplified 43.3%

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

9. Taylor expanded in eps around 0 50.8%

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

10. Final simplification 50.8%

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

Alternative 7: 44.9% 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 56.7%

   \[x - \sqrt{x \cdot x - \varepsilon} \]

2. Taylor expanded in x around inf 49.6%

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

3. Final simplification 49.6%

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

Alternative 8: 7.9% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 56.7%

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

   1. flip-- 56.7%

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

   2. div-inv 56.5%

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

   3. add-sqr-sqrt 56.3%

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

   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 74.9%

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

   9. hypot-def 74.9%

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

3. Applied egg-rr 74.9%

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

   1. unpow2 74.9%

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

5. Applied egg-rr 74.9%

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

   1. unpow2 74.9%

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

7. Applied egg-rr 74.9%

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

   1. un-div-inv 75.0%

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

   2. clear-num 74.8%

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

   3. add-sqr-sqrt 74.8%

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

   4. sqrt-unprod 76.8%

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

   5. sqr-neg 76.8%

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

   6. sqrt-prod 23.3%

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

   7. add-sqr-sqrt 23.3%

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

   8. +-inverses 23.3%

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

   9. +-lft-identity 23.3%

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

9. Applied egg-rr 23.3%

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

11. Simplified 7.8%

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

12. Final simplification 7.8%

    \[\leadsto \varepsilon \]

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023310 
(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))))