ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.4% → 99.9%

Time: 6.8s

Alternatives: 6

Speedup: 0.7×

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: 62.4% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - \varepsilon\\ \frac{\varepsilon}{\frac{t\_0}{\sqrt{t\_0}} + x} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* x x) eps))) (/ eps (+ (/ t_0 (sqrt t_0)) x))))

C

double code(double x, double eps) {
	double t_0 = (x * x) - eps;
	return eps / ((t_0 / sqrt(t_0)) + x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (x * x) - eps
    code = eps / ((t_0 / sqrt(t_0)) + x)
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x * x) - eps;
	return eps / ((t_0 / Math.sqrt(t_0)) + x);
}

Python

def code(x, eps):
	t_0 = (x * x) - eps
	return eps / ((t_0 / math.sqrt(t_0)) + x)

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) - eps)
	return Float64(eps / Float64(Float64(t_0 / sqrt(t_0)) + x))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (x * x) - eps;
	tmp = eps / ((t_0 / sqrt(t_0)) + x);
end

Wolfram

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

Derivation

1. Initial program 60.6%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-div N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower-/.f64 60.2

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

4. Applied rewrites 60.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. associate--r- N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 65.3

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

6. Applied rewrites 65.3%

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

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{x \cdot x - x \cdot x}{\sqrt{x \cdot x - \varepsilon} + x} + \frac{\varepsilon}{\sqrt{x \cdot x - \varepsilon} + x}} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-div N/A

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

   8. metadata-eval N/A

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

   9. unpow1 N/A

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

   10. pow1/2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift--.f64 N/A

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

   16. lift-sqrt.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \frac{\varepsilon}{\frac{x \cdot x - \varepsilon}{\sqrt{x \cdot x - \varepsilon}} + x} \]
12. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154) t_0 (* 0.5 (/ eps x)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-154)) then
        tmp = t_0
    else
        tmp = 0.5d0 * (eps / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-154:
		tmp = t_0
	else:
		tmp = 0.5 * (eps / x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(eps / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = 0.5 * (eps / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-154], t$95$0, N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.7%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 97.1

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

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{0.5}{x} \cdot \varepsilon} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.4%

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

Alternative 3: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-154)
   (- x (sqrt (- eps)))
   (* 0.5 (/ eps x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-1d-154)) then
        tmp = x - sqrt(-eps)
    else
        tmp = 0.5d0 * (eps / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-154) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-154:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = 0.5 * (eps / x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-154)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(0.5 * Float64(eps / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154)
		tmp = x - sqrt(-eps);
	else
		tmp = 0.5 * (eps / 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], -1e-154], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 97.1

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

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{0.5}{x} \cdot \varepsilon} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.4%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-div N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower-/.f64 60.2

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

4. Applied rewrites 60.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. associate--r- N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 65.3

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

6. Applied rewrites 65.3%

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

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{x \cdot x - x \cdot x}{\sqrt{x \cdot x - \varepsilon} + x} + \frac{\varepsilon}{\sqrt{x \cdot x - \varepsilon} + x}} \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-add-rev N/A

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

   5. lift--.f64 N/A

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

   6. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0} + \varepsilon}{\sqrt{x \cdot x - \varepsilon} + x} \]

   7. +-lft-identity N/A

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

   8. lift-/.f64 99.5

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

10. Applied rewrites 99.5%

    \[\leadsto \color{blue}{\frac{\varepsilon}{\sqrt{x \cdot x - \varepsilon} + x}} \]
11. Add Preprocessing

Alternative 5: 57.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 56.0

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

5. Applied rewrites 56.0%

   \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]
6. Add Preprocessing

Alternative 6: 3.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x - \left(-x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. lower-neg.f64 3.6

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

5. Applied rewrites 3.6%

   \[\leadsto x - \color{blue}{\left(-x\right)} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.7× 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 2024344 
(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))))