ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.6% → 99.5%

Time: 9.5s

Alternatives: 6

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: 62.6% 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 65.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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. rem-square-sqrt N/A

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

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

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

   11. rem-square-sqrt N/A

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

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

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

   13. *-lowering-*.f64 64.8%

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

4. Applied egg-rr 64.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.5%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-152) t_0 (* eps (/ 1.0 (+ x (+ x (/ (* eps -0.5) x))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-152)) then
        tmp = t_0
    else
        tmp = eps * (1.0d0 / (x + (x + ((eps * (-0.5d0)) / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-152:
		tmp = t_0
	else:
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-152)
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(1.0 / Float64(x + Float64(x + Float64(Float64(eps * -0.5) / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-152)
		tmp = t_0;
	else
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-152], t$95$0, N[(eps * N[(1.0 / N[(x + N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000007e-152

   1. Initial program 98.3%

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

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.0%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 7.2%

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

   5. Simplified 7.2%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

   7. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{\left(x \cdot x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right) \cdot \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\right) \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\varepsilon} \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-113}:\\ \;\;\;\;x - \sqrt{0 - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 7.8e-113)
   (- x (sqrt (- 0.0 eps)))
   (* eps (/ 1.0 (+ x (+ x (/ (* eps -0.5) x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7.8e-113) {
		tmp = x - sqrt((0.0 - eps));
	} else {
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.8d-113) then
        tmp = x - sqrt((0.0d0 - eps))
    else
        tmp = eps * (1.0d0 / (x + (x + ((eps * (-0.5d0)) / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.8e-113) {
		tmp = x - Math.sqrt((0.0 - eps));
	} else {
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 7.8e-113:
		tmp = x - math.sqrt((0.0 - eps))
	else:
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7.8e-113)
		tmp = Float64(x - sqrt(Float64(0.0 - eps)));
	else
		tmp = Float64(eps * Float64(1.0 / Float64(x + Float64(x + Float64(Float64(eps * -0.5) / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.8e-113)
		tmp = x - sqrt((0.0 - eps));
	else
		tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7.8e-113], N[(x - N[Sqrt[N[(0.0 - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(eps * N[(1.0 / N[(x + N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.7999999999999997e-113

   1. Initial program 95.6%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 94.7%

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

   5. Simplified 94.7%

      \[\leadsto x - \sqrt{\color{blue}{0 - \varepsilon}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 94.7%

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

   7. Applied egg-rr 94.7%

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

   if 7.7999999999999997e-113 < x

   1. Initial program 27.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 8.8%

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

   5. Simplified 8.8%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

   7. Applied egg-rr 8.8%

      \[\leadsto \color{blue}{\left(x \cdot x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right) \cdot \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\right) \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Simplified 80.8%

       \[\leadsto \color{blue}{\varepsilon} \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-113}:\\ \;\;\;\;x - \sqrt{0 - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (/ 1.0 (+ x (+ x (/ (* eps -0.5) x))))))

C

double code(double x, double eps) {
	return eps * (1.0 / (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 * (1.0d0 / (x + (x + ((eps * (-0.5d0)) / x))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
}

Python

def code(x, eps):
	return eps * (1.0 / (x + (x + ((eps * -0.5) / x))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 / Float64(x + Float64(x + Float64(Float64(eps * -0.5) / x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 / (x + (x + ((eps * -0.5) / x))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 / N[(x + N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 6.3%

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

5. Simplified 6.3%

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

   1. flip-- N/A

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

   2. div-inv N/A

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

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

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

7. Applied egg-rr 7.2%

   \[\leadsto \color{blue}{\left(x \cdot x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right) \cdot \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\right) \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}} \]

8. Taylor expanded in x around inf

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

10. Simplified 42.0%

    \[\leadsto \color{blue}{\varepsilon} \cdot \frac{1}{x + \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)} \]
11. Add Preprocessing

Alternative 5: 43.7% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in x around inf

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

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

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

   2. /-lowering-/.f64 41.3%

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

5. Simplified 41.3%

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

Alternative 6: 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 65.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 2024156 
(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))))