ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.9% → 99.5%

Time: 10.8s

Alternatives: 9

Speedup: 1.0×

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: 61.9% 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, 1.0× 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 56.6%

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

   1. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\left(x + \sqrt{x \cdot x - \varepsilon}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(\color{blue}{x} + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x - \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   12. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(x \cdot x - \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   14. *-lowering-*.f64 56.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 56.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.6%

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

   1. unpow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x - \varepsilon}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\left(x \cdot x - \varepsilon\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right)\right)\right) \]

   4. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right)\right)\right) \]

9. Applied egg-rr 99.6%

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

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon}{x}}{2 + -0.5 \cdot \frac{\varepsilon}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-154) t_0 (/ (/ eps x) (+ 2.0 (* -0.5 (/ eps (* x x))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-154) {
		tmp = t_0;
	} else {
		tmp = (eps / x) / (2.0 + (-0.5 * (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 <= (-4d-154)) then
        tmp = t_0
    else
        tmp = (eps / x) / (2.0d0 + ((-0.5d0) * (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 <= -4e-154) {
		tmp = t_0;
	} else {
		tmp = (eps / x) / (2.0 + (-0.5 * (eps / (x * x))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -4e-154:
		tmp = t_0
	else:
		tmp = (eps / x) / (2.0 + (-0.5 * (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 <= -4e-154)
		tmp = t_0;
	else
		tmp = Float64(Float64(eps / x) / Float64(2.0 + Float64(-0.5 * 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 <= -4e-154)
		tmp = t_0;
	else
		tmp = (eps / x) / (2.0 + (-0.5 * (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, -4e-154], t$95$0, N[(N[(eps / x), $MachinePrecision] / N[(2.0 + N[(-0.5 * N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -3.9999999999999999e-154

   1. Initial program 98.5%

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

   if -3.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.6%

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

      1. flip-- N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\left(x + \sqrt{x \cdot x - \varepsilon}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(\color{blue}{x} + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x - \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(x \cdot x - \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 7.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   4. Applied egg-rr 7.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left(x \cdot \left(2 + \frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(2 + \frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   10. Simplified 99.4%

       \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot \left(2 + \frac{\varepsilon \cdot -0.5}{x \cdot x}\right)}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\varepsilon}{x}\right), \color{blue}{\left(2 + \frac{\varepsilon \cdot \frac{-1}{2}}{x \cdot x}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \left(\color{blue}{2} + \frac{\varepsilon \cdot \frac{-1}{2}}{x \cdot x}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{\varepsilon \cdot \frac{-1}{2}}{x \cdot x}\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{x} \cdot x}\right)\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot \color{blue}{\frac{\varepsilon}{x \cdot x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{\varepsilon}{x \cdot x}\right)}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 99.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 99.4%

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

Alternative 3: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-123}:\\ \;\;\;\;x - \sqrt{0 - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon}{x}}{2 + -0.5 \cdot \frac{\varepsilon}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.8e-123)
   (- x (sqrt (- 0.0 eps)))
   (/ (/ eps x) (+ 2.0 (* -0.5 (/ eps (* x x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.8e-123) {
		tmp = x - sqrt((0.0 - eps));
	} else {
		tmp = (eps / x) / (2.0 + (-0.5 * (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 <= 2.8d-123) then
        tmp = x - sqrt((0.0d0 - eps))
    else
        tmp = (eps / x) / (2.0d0 + ((-0.5d0) * (eps / (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.8e-123) {
		tmp = x - Math.sqrt((0.0 - eps));
	} else {
		tmp = (eps / x) / (2.0 + (-0.5 * (eps / (x * x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.8e-123:
		tmp = x - math.sqrt((0.0 - eps))
	else:
		tmp = (eps / x) / (2.0 + (-0.5 * (eps / (x * x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.8e-123)
		tmp = Float64(x - sqrt(Float64(0.0 - eps)));
	else
		tmp = Float64(Float64(eps / x) / Float64(2.0 + Float64(-0.5 * Float64(eps / Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.8e-123)
		tmp = x - sqrt((0.0 - eps));
	else
		tmp = (eps / x) / (2.0 + (-0.5 * (eps / (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.8e-123], N[(x - N[Sqrt[N[(0.0 - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps / x), $MachinePrecision] / N[(2.0 + N[(-0.5 * N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999999e-123

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \varepsilon\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\left(0 - \varepsilon\right)\right)\right) \]

      3. --lowering--.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \varepsilon\right)\right)\right) \]

   5. Simplified 96.4%

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

   if 2.7999999999999999e-123 < x

   1. Initial program 27.4%

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

      1. flip-- N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\left(x + \sqrt{x \cdot x - \varepsilon}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(\color{blue}{x} + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x - \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(x \cdot x - \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 27.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   4. Applied egg-rr 27.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left(x \cdot \left(2 + \frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(2 + \frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   10. Simplified 80.1%

       \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot \left(2 + \frac{\varepsilon \cdot -0.5}{x \cdot x}\right)}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\varepsilon}{x}\right), \color{blue}{\left(2 + \frac{\varepsilon \cdot \frac{-1}{2}}{x \cdot x}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \left(\color{blue}{2} + \frac{\varepsilon \cdot \frac{-1}{2}}{x \cdot x}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{\varepsilon \cdot \frac{-1}{2}}{x \cdot x}\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{x} \cdot x}\right)\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot \color{blue}{\frac{\varepsilon}{x \cdot x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{\varepsilon}{x \cdot x}\right)}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 80.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, x\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 80.1%

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

Alternative 4: 45.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ (* x 2.0) (/ (* eps -0.5) x))))

C

double code(double x, double eps) {
	return eps / ((x * 2.0) + ((eps * -0.5) / x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / ((x * 2.0d0) + ((eps * (-0.5d0)) / x))
end function

Java

public static double code(double x, double eps) {
	return eps / ((x * 2.0) + ((eps * -0.5) / x));
}

Python

def code(x, eps):
	return eps / ((x * 2.0) + ((eps * -0.5) / x))

Julia

function code(x, eps)
	return Float64(eps / Float64(Float64(x * 2.0) + Float64(Float64(eps * -0.5) / x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / ((x * 2.0) + ((eps * -0.5) / x));
end

Wolfram

code[x_, eps_] := N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

   1. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\left(x + \sqrt{x \cdot x - \varepsilon}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(\color{blue}{x} + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x - \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   12. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(x \cdot x - \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   14. *-lowering-*.f64 56.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 56.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.6%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \frac{\varepsilon}{x} + 2 \cdot x\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \left(2 \cdot x + \color{blue}{\frac{-1}{2} \cdot \frac{\varepsilon}{x}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(2 \cdot x\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{\varepsilon}{x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{\varepsilon}{x}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{x}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{x}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), x\right)\right)\right) \]

   8. *-lowering-*.f64 50.4%

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), x\right)\right)\right) \]

10. Simplified 50.4%

    \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2 + \frac{\varepsilon \cdot -0.5}{x}}} \]
11. Add Preprocessing

Alternative 5: 45.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x (+ x (/ (* eps -0.5) x)))))

C

double code(double x, double eps) {
	return eps / (x + (x + ((eps * -0.5) / x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (x + (x + ((eps * (-0.5d0)) / x)))
end function

Java

public static double code(double x, double eps) {
	return eps / (x + (x + ((eps * -0.5) / x)));
}

Python

def code(x, eps):
	return eps / (x + (x + ((eps * -0.5) / x)))

Julia

function code(x, eps)
	return Float64(eps / Float64(x + Float64(x + Float64(Float64(eps * -0.5) / x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (x + (x + ((eps * -0.5) / x)));
end

Wolfram

code[x_, eps_] := N[(eps / N[(x + N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

   1. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\left(x + \sqrt{x \cdot x - \varepsilon}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(\color{blue}{x} + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x - \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   12. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(x \cdot x - \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   14. *-lowering-*.f64 56.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 56.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\varepsilon}, \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.6%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \color{blue}{\left(x + \frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)}\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{x}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), x\right)\right)\right)\right) \]

   5. *-lowering-*.f64 50.4%

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), x\right)\right)\right)\right) \]

10. Simplified 50.4%

    \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}} \]
11. Add Preprocessing

Alternative 6: 44.5% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot 0.5}{x} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (* eps 0.5) x))

C

double code(double x, double eps) {
	return (eps * 0.5) / x;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * 0.5d0) / x
end function

Java

public static double code(double x, double eps) {
	return (eps * 0.5) / x;
}

Python

def code(x, eps):
	return (eps * 0.5) / x

Julia

function code(x, eps)
	return Float64(Float64(eps * 0.5) / x)
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * 0.5) / x;
end

Wolfram

code[x_, eps_] := N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right), \color{blue}{x}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\varepsilon \cdot \frac{-1}{2}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\varepsilon \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right), x\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right), x\right) \]

   8. *-lowering-*.f64 49.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right), x\right) \]

5. Simplified 49.7%

   \[\leadsto \color{blue}{\frac{\varepsilon \cdot 0.5}{x}} \]
6. Add Preprocessing

Alternative 7: 44.3% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{x}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ 0.5 (/ x eps)))

C

double code(double x, double eps) {
	return 0.5 / (x / eps);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.5d0 / (x / eps)
end function

Java

public static double code(double x, double eps) {
	return 0.5 / (x / eps);
}

Python

def code(x, eps):
	return 0.5 / (x / eps)

Julia

function code(x, eps)
	return Float64(0.5 / Float64(x / eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 / (x / eps);
end

Wolfram

code[x_, eps_] := N[(0.5 / N[(x / eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + \frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{\varepsilon}{{x}^{2}}\right)}\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 6.2%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

5. Simplified 6.2%

   \[\leadsto x - \color{blue}{x \cdot \left(1 + \frac{\varepsilon \cdot -0.5}{x \cdot x}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate--r+ N/A

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

   4. +-inverses N/A

      \[\leadsto 0 - \color{blue}{\frac{\varepsilon \cdot \frac{-1}{2}}{x \cdot x}} \cdot x \]

   5. div-inv N/A

      \[\leadsto 0 - \left(\left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \frac{1}{x \cdot x}\right) \cdot x \]

   6. associate-*l* N/A

      \[\leadsto 0 - \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \color{blue}{\left(\frac{1}{x \cdot x} \cdot x\right)} \]

   7. pow2 N/A

      \[\leadsto 0 - \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \left(\frac{1}{{x}^{2}} \cdot x\right) \]

   8. pow-flip N/A

      \[\leadsto 0 - \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right) \]

   9. pow-plus N/A

      \[\leadsto 0 - \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot {x}^{\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)}} \]

   10. metadata-eval N/A

       \[\leadsto 0 - \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot {x}^{\left(-2 + 1\right)} \]

   11. metadata-eval N/A

       \[\leadsto 0 - \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot {x}^{-1} \]

   12. inv-pow N/A

       \[\leadsto 0 - \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \frac{1}{\color{blue}{x}} \]

   13. div-inv N/A

       \[\leadsto 0 - \frac{\varepsilon \cdot \frac{-1}{2}}{\color{blue}{x}} \]

   14. *-commutative N/A

       \[\leadsto 0 - \frac{\frac{-1}{2} \cdot \varepsilon}{x} \]

   15. associate-*r/ N/A

       \[\leadsto 0 - \frac{-1}{2} \cdot \color{blue}{\frac{\varepsilon}{x}} \]

   16. cancel-sign-sub-inv N/A

       \[\leadsto 0 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{\varepsilon}{x}} \]

   17. +-lft-identity N/A

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

   18. metadata-eval N/A

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

   19. clear-num N/A

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

   20. un-div-inv N/A

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

   21. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{x}{\varepsilon}\right)}\right) \]

   22. /-lowering-/.f64 49.5%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(x, \color{blue}{\varepsilon}\right)\right) \]

7. Applied egg-rr 49.5%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{x}{\varepsilon}}} \]
8. Add Preprocessing

Alternative 8: 5.4% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* x -2.0))

C

double code(double x, double eps) {
	return x * -2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return x * -2.0;
}

Python

def code(x, eps):
	return x * -2.0

Julia

function code(x, eps)
	return Float64(x * -2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = x * -2.0;
end

Wolfram

code[x_, eps_] := N[(x * -2.0), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

   1. flip-- N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\left(x + \sqrt{x \cdot x - \varepsilon}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(\color{blue}{x} + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x - \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \left(x + \sqrt{x \cdot x - \varepsilon}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot x - \varepsilon}\right)}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \left({\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right)}^{\frac{1}{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(\sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   12. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\left(x \cdot x - \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

   14. *-lowering-*.f64 56.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{1}{2}\right)\right)\right) \]

4. Applied egg-rr 56.3%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(x + \frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)}\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)}\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \left(\frac{\frac{-1}{2} \cdot \varepsilon}{\color{blue}{x}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{x}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), x\right)\right)\right)\right) \]

   5. *-lowering-*.f64 7.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), x\right)\right)\right)\right) \]

7. Simplified 7.3%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 5.5%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{-2}\right) \]

10. Simplified 5.5%

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

Alternative 9: 4.3% accurate, 107.0× 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 56.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{x}\right) \]
4. Step-by-step derivation

5. Simplified 4.2%

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

   1. +-inverses 4.2%

      \[\leadsto 0 \]

7. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.5% accurate, 1.0× 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 2024149 
(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))))