ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.2% → 98.6%

Time: 9.2s

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: 62.2% 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: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x * (2.0 + ((eps * -0.5) / pow(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-153)) then
        tmp = t_0
    else
        tmp = eps / (x * (2.0d0 + ((eps * (-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-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x * (2.0 + ((eps * -0.5) / Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-153:
		tmp = t_0
	else:
		tmp = eps / (x * (2.0 + ((eps * -0.5) / math.pow(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-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x * Float64(2.0 + Float64(Float64(eps * -0.5) / (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-153)
		tmp = t_0;
	else
		tmp = eps / (x * (2.0 + ((eps * -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-153], t$95$0, N[(eps / N[(x * N[(2.0 + N[(N[(eps * -0.5), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

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

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. associate--r- 99.6%

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

      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 49.8%

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

      9. hypot-define 49.8%

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

   4. Applied egg-rr 49.8%

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

      1. *-commutative 49.8%

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

      2. +-inverses 49.8%

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

      3. +-lft-identity 49.8%

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

      4. associate-*l/ 50.0%

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

      5. *-lft-identity 50.0%

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

   6. Simplified 50.0%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot \left(2 + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{{x}^{2}}\right)}} \]
   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. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 99.1%

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

      6. mul-1-neg 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. distribute-rgt-neg-in 99.1%

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

      9. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

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

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 7.5%

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

      4. associate--r- 99.6%

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

      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 49.8%

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

      9. hypot-define 49.8%

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

   4. Applied egg-rr 49.8%

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

      1. *-commutative 49.8%

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

      2. +-inverses 49.8%

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

      3. +-lft-identity 49.8%

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

      4. associate-*l/ 50.0%

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

      5. *-lft-identity 50.0%

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

   6. Simplified 50.0%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 99.1%

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

      6. mul-1-neg 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. distribute-rgt-neg-in 99.1%

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

      9. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

Alternative 3: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-93}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.7e-93)
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (/ (* eps -0.5) x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.7e-93) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((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 <= 3.7d-93) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + ((eps * (-0.5d0)) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.7e-93) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((eps * -0.5) / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.7e-93:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + ((eps * -0.5) / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.7e-93)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(eps * -0.5) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.7e-93)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + ((eps * -0.5) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.7e-93], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.70000000000000002e-93

   1. Initial program 92.9%

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

   3. Taylor expanded in x around 0 90.5%

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

      1. neg-mul-1 90.5%

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

   5. Simplified 90.5%

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

   if 3.70000000000000002e-93 < x

   1. Initial program 25.1%

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

      1. flip-- 25.1%

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

      2. div-inv 25.0%

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

      3. add-sqr-sqrt 25.1%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 60.0%

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

      9. hypot-define 60.0%

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

   4. Applied egg-rr 60.0%

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

      1. *-commutative 60.0%

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

      2. +-inverses 60.0%

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

      3. +-lft-identity 60.0%

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

      4. associate-*l/ 60.2%

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

      5. *-lft-identity 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in eps around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 82.3%

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

      6. mul-1-neg 82.3%

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

      7. distribute-lft-neg-in 82.3%

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

      8. distribute-rgt-neg-in 82.3%

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

      9. metadata-eval 82.3%

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

   9. Simplified 82.3%

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

Alternative 4: 44.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. flip-- 63.4%

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

   2. div-inv 63.2%

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

   3. add-sqr-sqrt 63.1%

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

   4. associate--r- 99.4%

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

   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 79.9%

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

   9. hypot-define 79.9%

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

4. Applied egg-rr 79.9%

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

   1. *-commutative 79.9%

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

   2. +-inverses 79.9%

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

   3. +-lft-identity 79.9%

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

   4. associate-*l/ 80.0%

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

   5. *-lft-identity 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in eps around 0 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 43.3%

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

   6. mul-1-neg 43.3%

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

   7. distribute-lft-neg-in 43.3%

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

   8. distribute-rgt-neg-in 43.3%

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

   9. metadata-eval 43.3%

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

9. Simplified 43.3%

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

Alternative 5: 44.1% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in x around inf 42.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]
4. 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 63.5%

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

   1. flip-- 63.4%

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

   2. div-inv 63.2%

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

   3. add-sqr-sqrt 63.1%

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

   4. associate--r- 99.4%

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

   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 79.9%

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

   9. hypot-define 79.9%

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

4. Applied egg-rr 79.9%

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

   1. *-commutative 79.9%

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

   2. +-inverses 79.9%

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

   3. +-lft-identity 79.9%

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

   4. associate-*l/ 80.0%

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

   5. *-lft-identity 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in eps around 0 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 43.3%

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

   6. mul-1-neg 43.3%

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

   7. distribute-lft-neg-in 43.3%

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

   8. distribute-rgt-neg-in 43.3%

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

   9. metadata-eval 43.3%

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

9. Simplified 43.3%

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

10. Taylor expanded in eps around inf 5.5%

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

    1. *-commutative 5.5%

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

12. Simplified 5.5%

    \[\leadsto \color{blue}{x \cdot -2} \]
13. 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 63.5%

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

3. Taylor expanded in x around inf 4.2%

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

4. Taylor expanded in x around 0 4.2%

   \[\leadsto \color{blue}{0} \]
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 2024181 
(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))))