ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.1% → 99.5%

Time: 7.2s

Alternatives: 7

Speedup: 0.7×

Specification

\[\left(0 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}

Python

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}

Python

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / (math.sqrt(((x * x) - eps)) + x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

4. Applied rewrites 49.8%

   \[\leadsto \color{blue}{\frac{{x}^{3} - {\left(x \cdot x - \varepsilon\right)}^{1.5}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{x}^{3} - {\left(x \cdot x - \varepsilon\right)}^{\frac{3}{2}}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{{x}^{3} - {\left(x \cdot x - \varepsilon\right)}^{\frac{3}{2}}}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)} \]

   3. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{x}^{3}} - {\left(x \cdot x - \varepsilon\right)}^{\frac{3}{2}}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{{x}^{3} - \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\frac{3}{2}}}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{{x}^{3} - {\left(x \cdot x - \varepsilon\right)}^{\color{blue}{\left(\frac{3}{2}\right)}}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)} \]

   6. sqrt-pow2 N/A

      \[\leadsto \frac{{x}^{3} - \color{blue}{{\left(\sqrt{x \cdot x - \varepsilon}\right)}^{3}}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{{x}^{3} - {\left(\sqrt{\color{blue}{x \cdot x - \varepsilon}}\right)}^{3}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{{x}^{3} - {\left(\sqrt{\color{blue}{x \cdot x} - \varepsilon}\right)}^{3}}{\mathsf{fma}\left(\sqrt{x \cdot x - \varepsilon}, x, \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)\right)} \]

   9. lift-fma.f64 N/A

      \[\leadsto \frac{{x}^{3} - {\left(\sqrt{x \cdot x - \varepsilon}\right)}^{3}}{\color{blue}{\sqrt{x \cdot x - \varepsilon} \cdot x + \mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right)}} \]

   10. +-commutative N/A

       \[\leadsto \frac{{x}^{3} - {\left(\sqrt{x \cdot x - \varepsilon}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(x, x, x \cdot x - \varepsilon\right) + \sqrt{x \cdot x - \varepsilon} \cdot x}} \]

   11. lift-fma.f64 N/A

       \[\leadsto \frac{{x}^{3} - {\left(\sqrt{x \cdot x - \varepsilon}\right)}^{3}}{\color{blue}{\left(x \cdot x + \left(x \cdot x - \varepsilon\right)\right)} + \sqrt{x \cdot x - \varepsilon} \cdot x} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{{x}^{3} - {\left(\sqrt{x \cdot x - \varepsilon}\right)}^{3}}{\left(\color{blue}{x \cdot x} + \left(x \cdot x - \varepsilon\right)\right) + \sqrt{x \cdot x - \varepsilon} \cdot x} \]

   13. associate-+r+ N/A

       \[\leadsto \frac{{x}^{3} - {\left(\sqrt{x \cdot x - \varepsilon}\right)}^{3}}{\color{blue}{x \cdot x + \left(\left(x \cdot x - \varepsilon\right) + \sqrt{x \cdot x - \varepsilon} \cdot x\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{{x}^{3} - {\left(\sqrt{x \cdot x - \varepsilon}\right)}^{3}}{\color{blue}{x \cdot x} + \left(\left(x \cdot x - \varepsilon\right) + \sqrt{x \cdot x - \varepsilon} \cdot x\right)} \]

6. Applied rewrites 62.4%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. +-inverses 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-153) t_0 (* (/ eps x) 0.5))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-153) {
		tmp = t_0;
	} else {
		tmp = (eps / x) * 0.5;
	}
	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-153)) then
        tmp = t_0
    else
        tmp = (eps / x) * 0.5d0
    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-153) {
		tmp = t_0;
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -4e-153:
		tmp = t_0
	else:
		tmp = (eps / x) * 0.5
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -4e-153)
		tmp = t_0;
	else
		tmp = Float64(Float64(eps / x) * 0.5);
	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-153)
		tmp = t_0;
	else
		tmp = (eps / x) * 0.5;
	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-153], t$95$0, N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 98.3%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.9%

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

   3. Taylor expanded in eps around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-153)
   (- x (sqrt (fma x x (- eps))))
   (* (/ eps x) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - sqrt(fma(x, x, -eps));
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -4e-153)
		tmp = Float64(x - sqrt(fma(x, x, Float64(-eps))));
	else
		tmp = Float64(Float64(eps / x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -4e-153], N[(x - N[Sqrt[N[(x * x + (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 98.3%

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

   3. Taylor expanded in eps around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \sqrt{-1 \cdot \varepsilon + \color{blue}{{x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.3%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.9%

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

   3. Taylor expanded in eps around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-153)
   (- x (sqrt (- eps)))
   (* (/ eps x) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-4d-153)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (eps / x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -4e-153:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (eps / x) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -4e-153)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(eps / x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153)
		tmp = x - sqrt(-eps);
	else
		tmp = (eps / x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -4e-153], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.9%

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

   3. Taylor expanded in eps around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

Alternative 5: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-153)
   (- x (sqrt (- eps)))
   (* (/ 0.5 x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-4d-153)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (0.5d0 / x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -4e-153:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (0.5 / x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -4e-153)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(0.5 / x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153)
		tmp = x - sqrt(-eps);
	else
		tmp = (0.5 / x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -4e-153], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 6: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-153) (- x (sqrt (- eps))) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-4d-153)) then
        tmp = x - sqrt(-eps)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -4e-153) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -4e-153:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -4e-153)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -4e-153)
		tmp = x - sqrt(-eps);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -4e-153], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000016e-153

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -4.00000000000000016e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.9%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. sqr-pow N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. unpow1 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. pow2 N/A

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

      12. sqrt-pow1 N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

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

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

      16. sqr-pow N/A

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

      17. pow1/2 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. distribute-neg-in N/A

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

      20. +-commutative N/A

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

      21. distribute-neg-in N/A

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

   4. Applied rewrites 6.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x - \varepsilon\right)}^{0.25}, -{\left(x \cdot x - \varepsilon\right)}^{0.25}, x\right)} \]

   5. Taylor expanded in eps around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft 5.1

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

   7. Applied rewrites 5.1%

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

Alternative 7: 4.3% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 62.6%

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

   1. lift--.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. sqr-pow N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. unpow1 N/A

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

   7. metadata-eval N/A

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

   8. sqrt-pow1 N/A

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

   9. pow2 N/A

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

   10. sqr-neg-rev N/A

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

   11. pow2 N/A

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

   12. sqrt-pow1 N/A

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

   13. metadata-eval N/A

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

   14. unpow1 N/A

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

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

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

   16. sqr-pow N/A

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

   17. pow1/2 N/A

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

   18. lift-sqrt.f64 N/A

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

   19. distribute-neg-in N/A

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

   20. +-commutative N/A

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

   21. distribute-neg-in N/A

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

4. Applied rewrites 61.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x - \varepsilon\right)}^{0.25}, -{\left(x \cdot x - \varepsilon\right)}^{0.25}, x\right)} \]

5. Taylor expanded in eps around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft 4.3

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

7. Applied rewrites 4.3%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / (x + math.sqrt(((x * x) - eps)))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024329 
(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))))