ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.3% → 99.5%

Time: 7.0s

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: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% 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]

Derivation

1. Initial program 60.1%

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

   1. add-sqr-sqrt 59.5%

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

   2. pow2 59.5%

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

   3. pow1/2 59.5%

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

   4. sqrt-pow1 59.5%

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

   5. metadata-eval 59.5%

      \[\leadsto x - {\left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.25}}\right)}^{2} \]

3. Applied egg-rr 59.5%

   \[\leadsto x - \color{blue}{{\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}} \]
4. Step-by-step derivation

   1. flip-- 59.5%

      \[\leadsto \color{blue}{\frac{x \cdot x - {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2} \cdot {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}}{x + {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}}} \]

   2. pow-pow 59.9%

      \[\leadsto \frac{x \cdot x - \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}}{x + {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   3. metadata-eval 59.9%

      \[\leadsto \frac{x \cdot x - {\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.5}} \cdot {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}}{x + {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   4. pow-pow 59.7%

      \[\leadsto \frac{x \cdot x - {\left(x \cdot x - \varepsilon\right)}^{0.5} \cdot \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}}}{x + {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   5. metadata-eval 59.7%

      \[\leadsto \frac{x \cdot x - {\left(x \cdot x - \varepsilon\right)}^{0.5} \cdot {\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.5}}}{x + {\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}} \]

   6. pow-pow 60.1%

      \[\leadsto \frac{x \cdot x - {\left(x \cdot x - \varepsilon\right)}^{0.5} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.5}}{x + \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(0.25 \cdot 2\right)}}} \]

   7. metadata-eval 60.1%

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

5. Applied egg-rr 60.1%

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

   1. pow-sqr 59.8%

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

   2. metadata-eval 59.8%

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

   3. unpow1 59.8%

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

   4. unpow1/2 59.8%

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

7. Simplified 59.8%

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

8. Taylor expanded in x around 0 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-152) t_0 (/ eps (+ x (+ x (* (/ eps x) -0.5)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-152) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-5d-152)) then
        tmp = t_0
    else
        tmp = eps / (x + (x + ((eps / x) * (-0.5d0))))
    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 <= -5e-152) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-152:
		tmp = t_0
	else:
		tmp = eps / (x + (x + ((eps / x) * -0.5)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-152)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(eps / x) * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -5e-152)
		tmp = t_0;
	else
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	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, -5e-152], t$95$0, N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.9999999999999997e-152

   1. Initial program 98.9%

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

   if -4.9999999999999997e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.6%

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

      1. flip-- 9.5%

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

      2. div-inv 9.5%

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

      3. add-sqr-sqrt 9.7%

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

      4. sub-neg 9.7%

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

      5. add-sqr-sqrt 2.3%

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

      6. hypot-def 2.3%

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

   3. Applied egg-rr 2.3%

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

      1. associate-*r/ 2.3%

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

      2. *-rgt-identity 2.3%

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

      3. associate--r- 45.9%

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

      4. +-inverses 45.9%

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

      5. +-lft-identity 45.9%

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 98.4%

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

      4. *-commutative 98.4%

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

      5. associate-*r* 98.4%

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

      6. metadata-eval 98.4%

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

      7. associate-*r/ 98.4%

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

      8. *-commutative 98.4%

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

      9. fma-def 98.4%

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

   8. Simplified 98.4%

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

      1. fma-udef 98.4%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \]

Alternative 3: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-112}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.9e-112)
   (- x (sqrt (- eps)))
   (/ eps (+ (* (/ eps x) -0.5) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.9e-112) {
		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 <= 3.9d-112) 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 <= 3.9e-112) {
		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 <= 3.9e-112:
		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 <= 3.9e-112)
		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 <= 3.9e-112)
		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, 3.9e-112], 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 < 3.9000000000000001e-112

   1. Initial program 96.9%

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

   2. Taylor expanded in x around 0 95.1%

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

      1. neg-mul-1 95.1%

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

   4. Simplified 95.1%

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

   if 3.9000000000000001e-112 < x

   1. Initial program 29.7%

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

      1. flip-- 29.7%

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

      2. div-inv 29.6%

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

      3. add-sqr-sqrt 29.7%

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

      4. sub-neg 29.7%

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

      5. add-sqr-sqrt 24.8%

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

      6. hypot-def 24.8%

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

   3. Applied egg-rr 24.8%

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

      1. associate-*r/ 24.8%

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

      2. *-rgt-identity 24.8%

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

      3. associate--r- 59.2%

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

      4. +-inverses 59.2%

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

      5. +-lft-identity 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-*r* 79.5%

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

      6. metadata-eval 79.5%

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

      7. associate-*r/ 79.5%

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

      8. *-commutative 79.5%

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

      9. fma-def 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in x around 0 79.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-112}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\varepsilon}{x} \cdot -0.5 + x \cdot 2}\\ \end{array} \]

Alternative 4: 45.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.1%

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

   1. flip-- 60.1%

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

   2. div-inv 59.9%

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

   3. add-sqr-sqrt 59.8%

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

   4. sub-neg 59.8%

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

   5. add-sqr-sqrt 56.6%

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

   6. hypot-def 56.6%

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

3. Applied egg-rr 56.6%

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

   1. associate-*r/ 56.6%

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

   2. *-rgt-identity 56.6%

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

   3. associate--r- 76.1%

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

   4. +-inverses 76.1%

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

   5. +-lft-identity 76.1%

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

5. Simplified 76.1%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 47.6%

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

   4. *-commutative 47.6%

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

   5. associate-*r* 47.6%

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

   6. metadata-eval 47.6%

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

   7. associate-*r/ 47.6%

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

   8. *-commutative 47.6%

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

   9. fma-def 47.6%

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

8. Simplified 47.6%

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

   1. fma-udef 47.6%

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

10. Applied egg-rr 47.6%

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

11. Final simplification 47.6%

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

Alternative 5: 45.8% 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 60.1%

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

   1. flip-- 60.1%

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

   2. div-inv 59.9%

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

   3. add-sqr-sqrt 59.8%

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

   4. sub-neg 59.8%

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

   5. add-sqr-sqrt 56.6%

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

   6. hypot-def 56.6%

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

3. Applied egg-rr 56.6%

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

   1. associate-*r/ 56.6%

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

   2. *-rgt-identity 56.6%

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

   3. associate--r- 76.1%

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

   4. +-inverses 76.1%

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

   5. +-lft-identity 76.1%

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

5. Simplified 76.1%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 47.6%

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

   4. *-commutative 47.6%

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

   5. associate-*r* 47.6%

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

   6. metadata-eval 47.6%

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

   7. associate-*r/ 47.6%

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

   8. *-commutative 47.6%

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

   9. fma-def 47.6%

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

8. Simplified 47.6%

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

9. Taylor expanded in x around 0 47.6%

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

10. Final simplification 47.6%

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

Alternative 6: 45.0% 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 60.1%

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

2. Taylor expanded in x around inf 46.5%

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

3. Final simplification 46.5%

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

Alternative 7: 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 60.1%

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

   1. flip-- 60.1%

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

   2. div-inv 59.9%

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

   3. add-sqr-sqrt 59.8%

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

   4. sub-neg 59.8%

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

   5. add-sqr-sqrt 56.6%

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

   6. hypot-def 56.6%

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

3. Applied egg-rr 56.6%

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

   1. associate-*r/ 56.6%

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

   2. *-rgt-identity 56.6%

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

   3. associate--r- 76.1%

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

   4. +-inverses 76.1%

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

   5. +-lft-identity 76.1%

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

5. Simplified 76.1%

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

6. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 47.6%

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

   4. *-commutative 47.6%

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

   5. associate-*r* 47.6%

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

   6. metadata-eval 47.6%

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

   7. associate-*r/ 47.6%

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

   8. *-commutative 47.6%

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

   9. fma-def 47.6%

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

8. Simplified 47.6%

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

9. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

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

11. Simplified 5.3%

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

12. Final simplification 5.3%

    \[\leadsto x \cdot -2 \]

Alternative 8: 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 60.1%

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

   1. add-sqr-sqrt 59.5%

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

   2. pow2 59.5%

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

   3. pow1/2 59.5%

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

   4. sqrt-pow1 59.5%

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

   5. metadata-eval 59.5%

      \[\leadsto x - {\left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.25}}\right)}^{2} \]

3. Applied egg-rr 59.5%

   \[\leadsto x - \color{blue}{{\left({\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}^{2}} \]

4. Taylor expanded in x around inf 4.2%

   \[\leadsto \color{blue}{-1 \cdot x + x} \]
5. Step-by-step derivation

   1. distribute-lft1-in 4.2%

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

   2. metadata-eval 4.2%

      \[\leadsto \color{blue}{0} \cdot x \]

   3. mul0-lft 4.2%

      \[\leadsto \color{blue}{0} \]

6. Simplified 4.2%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 4.2%

   \[\leadsto 0 \]

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 2023240 
(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))))