ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.4% → 99.5%

Time: 10.3s

Alternatives: 7

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: 61.4% 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

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

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 56.0%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

4. Applied rewrites 55.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. +-inverses N/A

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

   4. +-rgt-identity 99.6

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

8. Applied rewrites 99.6%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(-0.5, \frac{\varepsilon}{x}, x\right) + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153) t_0 (/ eps (+ (fma -0.5 (/ eps x) x) x)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = eps / (fma(-0.5, (eps / x), 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-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(fma(-0.5, Float64(eps / x), x) + x));
	end
	return 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-153], t$95$0, N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 99.2%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 7.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. +-inverses N/A

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

      4. +-rgt-identity 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 99.7

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

   11. Applied rewrites 99.7%

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

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153) 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 <= -5e-153) {
		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 <= (-5d-153)) 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 <= -5e-153) {
		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 <= -5e-153:
		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 <= -5e-153)
		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 <= -5e-153)
		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, -5e-153], 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))) < -5.00000000000000033e-153

   1. Initial program 99.2%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 7.1%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (- eps)))
   (* 0.5 (/ eps x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		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))) <= (-5d-153)) 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))) <= -5e-153) {
		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))) <= -5e-153:
		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))) <= -5e-153)
		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))) <= -5e-153)
		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], -5e-153], 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))) < -5.00000000000000033e-153

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around inf

      \[\leadsto x - \sqrt{\color{blue}{-1 \cdot \varepsilon}} \]
   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 95.3

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

   5. Applied rewrites 95.3%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 7.1%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (- eps)))
   (* (/ 0.5 x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	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-153)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (0.5d0 / x) * eps
    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-153) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-153:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (0.5 / x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-153)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(0.5 / x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153)
		tmp = x - sqrt(-eps);
	else
		tmp = (0.5 / x) * eps;
	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-153], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around inf

      \[\leadsto x - \sqrt{\color{blue}{-1 \cdot \varepsilon}} \]
   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 95.3

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

   5. Applied rewrites 95.3%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 6: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;x - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2e-310) (- x (sqrt (- eps))) (- x (- x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2e-310) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = 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 (eps <= (-2d-310)) then
        tmp = x - sqrt(-eps)
    else
        tmp = x - -x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -2e-310) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = x - -x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -2e-310:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = x - -x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -2e-310)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(x - Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -2e-310)
		tmp = x - sqrt(-eps);
	else
		tmp = x - -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2e-310], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(x - (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.999999999999994e-310

   1. Initial program 72.9%

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

   3. Taylor expanded in eps around inf

      \[\leadsto x - \sqrt{\color{blue}{-1 \cdot \varepsilon}} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

   if -1.999999999999994e-310 < eps

   1. Initial program 8.4%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 6.1

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

   5. Applied rewrites 6.1%

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

Alternative 7: 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 56.0%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 2024277 
(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))))