ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.1% → 97.9%

Time: 10.0s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% 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: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-149}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot \left(2 + \frac{\varepsilon}{x} \cdot \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-149)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (* x (+ 2.0 (* (/ eps x) (/ -0.5 x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-149) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-149) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-149:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-149)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(x * Float64(2.0 + Float64(Float64(eps / x) * Float64(-0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-149)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-149], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x * N[(2.0 + N[(N[(eps / x), $MachinePrecision] * 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))) < -1.99999999999999996e-149

   1. Initial program 99.1%

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

      1. flip-- 99.0%

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

      2. div-inv 98.7%

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

      3. add-sqr-sqrt 98.5%

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

      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 -1.99999999999999996e-149 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.2%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.4%

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

      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 46.2%

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

      9. hypot-define 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. *-commutative 46.2%

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

      2. +-inverses 46.2%

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

      3. +-lft-identity 46.2%

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

      4. associate-*l/ 46.4%

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

      5. *-lft-identity 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. associate-*r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 99.3%

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

      6. associate-*l* 99.3%

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

      7. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. unpow2 99.3%

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

       2. times-frac 99.3%

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

   11. Applied egg-rr 99.3%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot \left(2 + \frac{\varepsilon}{x} \cdot \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-149) t_0 (/ eps (* x (+ 2.0 (* (/ eps x) (/ -0.5 x))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-149) {
		tmp = t_0;
	} else {
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-2d-149)) then
        tmp = t_0
    else
        tmp = eps / (x * (2.0d0 + ((eps / x) * ((-0.5d0) / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-149) {
		tmp = t_0;
	} else {
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-149:
		tmp = t_0
	else:
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -2e-149)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x * Float64(2.0 + Float64(Float64(eps / x) * Float64(-0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -2e-149)
		tmp = t_0;
	else
		tmp = eps / (x * (2.0 + ((eps / x) * (-0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-149], t$95$0, N[(eps / N[(x * N[(2.0 + N[(N[(eps / x), $MachinePrecision] * 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))) < -1.99999999999999996e-149

   1. Initial program 99.1%

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

   if -1.99999999999999996e-149 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.2%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.4%

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

      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 46.2%

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

      9. hypot-define 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. *-commutative 46.2%

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

      2. +-inverses 46.2%

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

      3. +-lft-identity 46.2%

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

      4. associate-*l/ 46.4%

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

      5. *-lft-identity 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. associate-*r* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 99.3%

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

      6. associate-*l* 99.3%

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

      7. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. unpow2 99.3%

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

       2. times-frac 99.3%

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

   11. Applied egg-rr 99.3%

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

Alternative 3: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-87}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5.6e-87)
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5.6e-87) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 5.6d-87) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / ((x * 2.0d0) + (eps * ((-0.5d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 5.6e-87) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 5.6e-87:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5.6e-87)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(eps * Float64(-0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 5.6e-87)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5.6e-87], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.6000000000000002e-87

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 84.8%

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

      1. neg-mul-1 84.8%

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

   5. Simplified 84.8%

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

   if 5.6000000000000002e-87 < x

   1. Initial program 17.4%

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

      1. flip-- 17.4%

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

      2. div-inv 17.4%

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

      3. add-sqr-sqrt 17.6%

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

      4. associate--r- 99.6%

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

      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 52.6%

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

      9. hypot-define 52.6%

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

   4. Applied egg-rr 52.6%

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

      1. *-commutative 52.6%

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

      2. +-inverses 52.6%

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

      3. +-lft-identity 52.6%

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

      4. associate-*l/ 52.7%

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

      5. *-lft-identity 52.7%

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

   6. Simplified 52.7%

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

   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. associate-*r* 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 90.7%

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

      8. associate-*l* 90.7%

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

      9. metadata-eval 90.7%

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

      10. associate-*r/ 90.7%

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

   9. Simplified 90.7%

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

Alternative 4: 45.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.9%

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

   1. flip-- 58.9%

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

   2. div-inv 58.7%

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

   3. add-sqr-sqrt 58.6%

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

   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 76.0%

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

   9. hypot-define 76.0%

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

4. Applied egg-rr 76.0%

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

   1. *-commutative 76.0%

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

   2. +-inverses 76.0%

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

   3. +-lft-identity 76.0%

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

   4. associate-*l/ 76.1%

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

   5. *-lft-identity 76.1%

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

6. Simplified 76.1%

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

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. associate-*r* 0.0%

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

   5. *-commutative 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 48.1%

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

   8. associate-*l* 48.1%

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

   9. metadata-eval 48.1%

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

   10. associate-*r/ 48.1%

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

9. Simplified 48.1%

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

Alternative 5: 44.3% 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 58.9%

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

3. Taylor expanded in x around inf 47.3%

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

4. Final simplification 47.3%

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

Alternative 6: 5.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* x -2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.9%

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

   1. flip-- 58.9%

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

   2. div-inv 58.7%

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

   3. add-sqr-sqrt 58.6%

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

   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 76.0%

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

   9. hypot-define 76.0%

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

4. Applied egg-rr 76.0%

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

   1. *-commutative 76.0%

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

   2. +-inverses 76.0%

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

   3. +-lft-identity 76.0%

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

   4. associate-*l/ 76.1%

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

   5. *-lft-identity 76.1%

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

6. Simplified 76.1%

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

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. associate-*r* 0.0%

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

   5. *-commutative 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 48.1%

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

   8. associate-*l* 48.1%

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

   9. metadata-eval 48.1%

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

   10. associate-*r/ 48.1%

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

9. Simplified 48.1%

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

10. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

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

12. Simplified 5.3%

    \[\leadsto \color{blue}{x \cdot -2} \]
13. Add Preprocessing

Alternative 7: 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 58.9%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 2024157 
(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))))