ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.2% → 99.5%

Time: 12.0s

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: 62.2% 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 63.7%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. associate--r+ N/A

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

   7. flip-- N/A

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

   8. sqr-neg N/A

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

   9. sub-neg N/A

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

   10. flip-+ N/A

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

   11. --lowering--.f64 N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. +-lowering-+.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. --lowering--.f64 N/A

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

   17. *-lowering-*.f64 63.4

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

4. Applied egg-rr 63.4%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.6%

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

Alternative 2: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{fma}\left(\varepsilon, \frac{-0.5}{x}, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-153) t_0 (/ eps (+ x (fma eps (/ -0.5 x) x))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x + fma(eps, (-0.5 / 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 <= -1e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + fma(eps, Float64(-0.5 / 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, -1e-153], t$95$0, N[(eps / N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000004e-153

   1. Initial program 99.4%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.1%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. flip-- N/A

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

      8. sqr-neg N/A

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

      9. sub-neg N/A

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

      10. flip-+ N/A

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

      11. --lowering--.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. *-lowering-*.f64 8.2

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

   4. Applied egg-rr 8.2%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 98.7

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

   10. Simplified 98.7%

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

Alternative 3: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-153) 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 <= -1e-153) {
		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 <= (-1d-153)) 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 <= -1e-153) {
		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 <= -1e-153:
		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 <= -1e-153)
		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 <= -1e-153)
		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, -1e-153], 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))) < -1.00000000000000004e-153

   1. Initial program 99.4%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.1%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. flip-- N/A

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

      8. sqr-neg N/A

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

      9. sub-neg N/A

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

      10. flip-+ N/A

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

      11. --lowering--.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. *-lowering-*.f64 8.2

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

   4. Applied egg-rr 8.2%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 97.9%

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

Alternative 4: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-153)
   (- x (sqrt (- eps)))
   (/ eps (+ x x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-153) {
		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))) <= (-1d-153)) 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))) <= -1e-153) {
		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))) <= -1e-153:
		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))) <= -1e-153)
		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))) <= -1e-153)
		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], -1e-153], 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))) < -1.00000000000000004e-153

   1. Initial program 99.4%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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. neg-lowering-neg.f64 96.8

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

   5. Simplified 96.8%

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

   if -1.00000000000000004e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.1%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. flip-- N/A

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

      8. sqr-neg N/A

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

      9. sub-neg N/A

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

      10. flip-+ N/A

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

      11. --lowering--.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. *-lowering-*.f64 8.2

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

   4. Applied egg-rr 8.2%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 97.9%

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

Alternative 5: 44.1% 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 63.7%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. associate--r+ N/A

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

   7. flip-- N/A

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

   8. sqr-neg N/A

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

   9. sub-neg N/A

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

   10. flip-+ N/A

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

   11. --lowering--.f64 N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. +-lowering-+.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. --lowering--.f64 N/A

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

   17. *-lowering-*.f64 63.4

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

4. Applied egg-rr 63.4%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.6%

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

8. Taylor expanded in x around inf

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

10. Simplified 42.3%

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

Alternative 6: 11.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. associate--r+ N/A

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

   7. flip-- N/A

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

   8. sqr-neg N/A

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

   9. sub-neg N/A

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

   10. flip-+ N/A

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

   11. --lowering--.f64 N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. +-lowering-+.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. --lowering--.f64 N/A

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

   17. *-lowering-*.f64 63.4

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

4. Applied egg-rr 63.4%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. +-lowering-+.f64 N/A

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

   13. +-inverses 99.4

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\frac{x + \sqrt{\color{blue}{-1 \cdot \varepsilon}}}{\varepsilon + 0}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{\frac{x + \sqrt{\color{blue}{\mathsf{neg}\left(\varepsilon\right)}}}{\varepsilon + 0}} \]

   2. neg-lowering-neg.f64 62.2

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

9. Simplified 62.2%

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

10. Taylor expanded in x around inf

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

    1. /-lowering-/.f64 10.9

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

12. Simplified 10.9%

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

Alternative 7: 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 63.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 4.3%

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

   1. +-inverses 4.3

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

7. Applied egg-rr 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 2024204 
(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))))