ENA, Section 1.4, Exercise 4d

Percentage Accurate: 63.0% → 99.5%

Time: 13.2s

Alternatives: 5

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: 63.0% 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 62.3%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. unpow1 N/A

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

   4. metadata-eval N/A

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

   5. sqr-pow N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. unpow1/2 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. unpow1/2 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-neg.f64 61.8

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

4. Applied rewrites 61.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\sqrt{x \cdot x - \varepsilon}\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-neg.f64 N/A

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

   6. sub-neg N/A

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

   7. flip-- N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-rgt-identity N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. rem-square-sqrt N/A

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

   14. sub-div N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\varepsilon + 0}{x + \sqrt{x \cdot x - \varepsilon}}} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-rgt-identity 99.4

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

8. Applied rewrites 99.4%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-155) t_0 (/ (* eps 0.5) x))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-155) {
		tmp = t_0;
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-5d-155)) then
        tmp = t_0
    else
        tmp = (eps * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-155) {
		tmp = t_0;
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-155:
		tmp = t_0
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-155)
		tmp = t_0;
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -5e-155)
		tmp = t_0;
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-155], t$95$0, N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.9999999999999999e-155

   1. Initial program 97.9%

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

   if -4.9999999999999999e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 10.1%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 3: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-155}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-155)
   (- x (sqrt (- eps)))
   (/ (* eps 0.5) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-155) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-5d-155)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (eps * 0.5d0) / 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-155) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-155:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-155)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-155)
		tmp = x - sqrt(-eps);
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-155], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.9999999999999999e-155

   1. Initial program 97.9%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 94.8

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

   5. Applied rewrites 94.8%

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

   if -4.9999999999999999e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 10.1%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 4: 59.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-155}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-155) (- x (sqrt (- eps))) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-155) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-5d-155)) then
        tmp = x - sqrt(-eps)
    else
        tmp = 0.0d0
    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-155) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-155:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-155)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-155)
		tmp = x - sqrt(-eps);
	else
		tmp = 0.0;
	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-155], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.9999999999999999e-155

   1. Initial program 97.9%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 94.8

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

   5. Applied rewrites 94.8%

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

   if -4.9999999999999999e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 10.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. sqr-pow N/A

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

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

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 9.4%

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

   5. Taylor expanded in eps around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft 5.2

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

   7. Applied rewrites 5.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 4.3% accurate, 22.0× 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 62.3%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. sqr-pow N/A

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

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

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in eps around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft 4.3

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

7. Applied rewrites 4.3%

   \[\leadsto \color{blue}{0} \]
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 2024238 
(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))))