ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.4% → 99.5%

Time: 12.7s

Alternatives: 8

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: 99.5% 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]

Derivation

1. Initial program 63.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. flip-+ N/A

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

   7. sqr-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lift-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower--.f64 N/A

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

4. Applied rewrites 62.9%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x - \frac{\varepsilon}{x + x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-154) t_0 (/ eps (+ x (- x (/ eps (+ x x))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x - (eps / (x + 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 <= (-5d-154)) then
        tmp = t_0
    else
        tmp = eps / (x + (x - (eps / (x + 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 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x - (eps / (x + x))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-154:
		tmp = t_0
	else:
		tmp = eps / (x + (x - (eps / (x + x))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x - Float64(eps / Float64(x + 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 <= -5e-154)
		tmp = t_0;
	else
		tmp = eps / (x + (x - (eps / (x + 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, -5e-154], t$95$0, N[(eps / N[(x + N[(x - N[(eps / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.0000000000000002e-154

   1. Initial program 98.2%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. sqr-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \frac{\mathsf{neg}\left(\varepsilon\right)}{\color{blue}{\left(\frac{1}{2} \cdot \frac{\varepsilon}{x} - x\right)} - x} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 98.3

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

   10. Applied rewrites 98.3%

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

       1. *-commutative N/A

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

       2. div-inv N/A

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

       3. associate-*r* N/A

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

       4. div-inv N/A

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

       5. metadata-eval N/A

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

       6. associate-/r* N/A

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

       7. count-2 N/A

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

       8. lift-+.f64 N/A

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

       9. /-rgt-identity N/A

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

       10. un-div-inv N/A

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

       11. /-rgt-identity N/A

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

       12. lower-/.f64 98.3

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

   12. Applied rewrites 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x - \frac{\varepsilon}{x + x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-154) t_0 (/ eps (+ x x)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + 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 <= (-5d-154)) then
        tmp = t_0
    else
        tmp = eps / (x + 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 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-154:
		tmp = t_0
	else:
		tmp = eps / (x + x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + 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 <= -5e-154)
		tmp = t_0;
	else
		tmp = eps / (x + 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, -5e-154], t$95$0, N[(eps / N[(x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.0000000000000002e-154

   1. Initial program 98.2%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.6%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 97.0

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

   7. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. count-2 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. /-rgt-identity N/A

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

      11. lower-/.f64 97.5

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

   9. Applied rewrites 97.5%

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

Alternative 4: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-154)
   (- x (sqrt (- eps)))
   (/ eps (+ x x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-154) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + 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))) <= (-5d-154)) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + 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))) <= -5e-154) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-154:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-154)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-154)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + 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], -5e-154], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.0000000000000002e-154

   1. Initial program 98.2%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.6%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 97.0

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

   7. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. count-2 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. /-rgt-identity N/A

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

      11. lower-/.f64 97.5

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

   9. Applied rewrites 97.5%

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

Alternative 5: 44.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / (x + x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 42.9

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

5. Applied rewrites 42.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 42.7

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

7. Applied rewrites 42.7%

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

   1. lift-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. count-2 N/A

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

   7. lift-+.f64 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. /-rgt-identity N/A

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

   11. lower-/.f64 42.9

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

9. Applied rewrites 42.9%

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

Alternative 6: 5.6% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ x x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * (x + x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 42.9

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

5. Applied rewrites 42.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 42.7

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

7. Applied rewrites 42.7%

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

   1. div-inv N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-/r* N/A

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

   5. count-2 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. flip-+ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. +-inverses N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{0}}{x - x}} \cdot \varepsilon \]

   12. +-inverses N/A

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

   13. +-inverses N/A

       \[\leadsto \frac{1}{\frac{x - x}{\color{blue}{0}}} \cdot \varepsilon \]

   14. +-inverses N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. clear-num N/A

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

   18. flip-+ N/A

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

   19. lift-+.f64 N/A

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

   20. lower-*.f64 5.5

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

9. Applied rewrites 5.5%

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

10. Final simplification 5.5%

    \[\leadsto \varepsilon \cdot \left(x + x\right) \]
11. Add Preprocessing

Alternative 7: 5.3% accurate, 3.7× 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 63.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. flip-+ N/A

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

   7. sqr-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lift-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower--.f64 N/A

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

4. Applied rewrites 62.9%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\mathsf{neg}\left(\varepsilon\right)}{\color{blue}{\left(\frac{1}{2} \cdot \frac{\varepsilon}{x} - x\right)} - x} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower--.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 43.5

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

10. Applied rewrites 43.5%

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

11. Taylor expanded in eps around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 5.0

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

13. Applied rewrites 5.0%

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

Alternative 8: 3.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ x + x \end{array} \]

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 + x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 3.7

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

5. Applied rewrites 3.7%

   \[\leadsto x - \color{blue}{\left(-x\right)} \]
6. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lower-+.f64 3.7

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

7. Applied rewrites 3.7%

   \[\leadsto \color{blue}{x + x} \]
8. 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 2024219 
(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))))