ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.9% → 98.3%

Time: 9.5s

Alternatives: 7

Speedup: 0.5×

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: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}, 0.125, 0.5 \cdot \varepsilon\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-152)
     t_0
     (/ (fma (* (/ eps x) (/ eps x)) 0.125 (* 0.5 eps)) x))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-152) {
		tmp = t_0;
	} else {
		tmp = fma(((eps / x) * (eps / x)), 0.125, (0.5 * eps)) / x;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -4e-152)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(eps / x) * Float64(eps / x)), 0.125, Float64(0.5 * eps)) / x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-152], t$95$0, N[(N[(N[(N[(eps / x), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000026e-152

   1. Initial program 98.6%

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

   if -4.00000000000000026e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-div N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. pow1/2 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-/.f64 7.2

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

   4. Applied rewrites 7.2%

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

   5. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 6.4

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

   7. Applied rewrites 6.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} - \frac{-1}{2} \cdot \varepsilon}{x}} \]

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

         \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \frac{{\varepsilon}^{2}}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{\varepsilon}^{2}}{{x}^{2}} \cdot \frac{1}{8}} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{{\varepsilon}^{2}}{{x}^{2}} \cdot \frac{1}{8} + \color{blue}{\frac{1}{2}} \cdot \varepsilon}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{{\varepsilon}^{2}}{{x}^{2}}, \frac{1}{8}, \frac{1}{2} \cdot \varepsilon\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\varepsilon \cdot \varepsilon}}{{x}^{2}}, \frac{1}{8}, \frac{1}{2} \cdot \varepsilon\right)}{x} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\varepsilon \cdot \varepsilon}{\color{blue}{x \cdot x}}, \frac{1}{8}, \frac{1}{2} \cdot \varepsilon\right)}{x} \]

      8. times-frac N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}}, \frac{1}{8}, \frac{1}{2} \cdot \varepsilon\right)}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}}, \frac{1}{8}, \frac{1}{2} \cdot \varepsilon\right)}{x} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\varepsilon}{x}} \cdot \frac{\varepsilon}{x}, \frac{1}{8}, \frac{1}{2} \cdot \varepsilon\right)}{x} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\varepsilon}{x} \cdot \color{blue}{\frac{\varepsilon}{x}}, \frac{1}{8}, \frac{1}{2} \cdot \varepsilon\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}, \frac{1}{8}, \color{blue}{\varepsilon \cdot \frac{1}{2}}\right)}{x} \]

      13. lower-*.f64 99.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}, 0.125, \color{blue}{\varepsilon \cdot 0.5}\right)}{x} \]

   10. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}, 0.125, \varepsilon \cdot 0.5\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\varepsilon}{x} \cdot \frac{\varepsilon}{x}, 0.125, 0.5 \cdot \varepsilon\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, 0.125, 0.5\right)}{x} \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-152) t_0 (* (/ (fma (/ eps (* x x)) 0.125 0.5) x) eps))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-152) {
		tmp = t_0;
	} else {
		tmp = (fma((eps / (x * x)), 0.125, 0.5) / x) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -4e-152)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(eps / Float64(x * x)), 0.125, 0.5) / x) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-152], t$95$0, N[(N[(N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.125 + 0.5), $MachinePrecision] / x), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000026e-152

   1. Initial program 98.6%

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

   if -4.00000000000000026e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.3%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{\varepsilon}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon} \]

      2. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{8} \cdot \frac{\color{blue}{1 \cdot \varepsilon}}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      3. associate-*l/ N/A

         \[\leadsto \left(\frac{1}{8} \cdot \color{blue}{\left(\frac{1}{{x}^{3}} \cdot \varepsilon\right)} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) \cdot \varepsilon} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\varepsilon \cdot \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) \cdot \varepsilon} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}, \varepsilon, \frac{1}{2} \cdot \frac{1}{x}\right)} \cdot \varepsilon \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{8} \cdot 1}{{x}^{3}}}, \varepsilon, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{8}}}{{x}^{3}}, \varepsilon, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{8}}{{x}^{3}}}, \varepsilon, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{8}}{\color{blue}{{x}^{3}}}, \varepsilon, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot \varepsilon \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, \varepsilon, \color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right) \cdot \varepsilon \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, \varepsilon, \frac{\color{blue}{\frac{1}{2}}}{x}\right) \cdot \varepsilon \]

      15. lower-/.f64 92.4

          \[\leadsto \mathsf{fma}\left(\frac{0.125}{{x}^{3}}, \varepsilon, \color{blue}{\frac{0.5}{x}}\right) \cdot \varepsilon \]

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{{x}^{3}}, \varepsilon, \frac{0.5}{x}\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{\varepsilon}{{x}^{2}}}{x} \cdot \varepsilon \]
   7. Step-by-step derivation

   8. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\varepsilon}{x}}{x}, 0.125, 0.5\right)}{x} \cdot \varepsilon \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{\varepsilon}{{x}^{2}}}{x} \cdot \varepsilon \]
   10. Step-by-step derivation

   11. Applied rewrites 98.8%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, 0.125, 0.5\right)}{x} \cdot \varepsilon \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -4e-152) t_0 (/ (* 0.5 eps) x))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-152) {
		tmp = t_0;
	} else {
		tmp = (0.5 * eps) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-4d-152)) then
        tmp = t_0
    else
        tmp = (0.5d0 * eps) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -4e-152) {
		tmp = t_0;
	} else {
		tmp = (0.5 * eps) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -4e-152:
		tmp = t_0
	else:
		tmp = (0.5 * eps) / x
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -4e-152)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * eps) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -4e-152)
		tmp = t_0;
	else
		tmp = (0.5 * eps) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-152], t$95$0, N[(N[(0.5 * eps), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000026e-152

   1. Initial program 98.6%

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

   if -4.00000000000000026e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{0.5}{x} \cdot \varepsilon} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.3%

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

Alternative 4: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-152)
   (- x (sqrt (- eps)))
   (/ (* 0.5 eps) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-152) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (0.5 * eps) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-4d-152)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (0.5d0 * eps) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -4e-152) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (0.5 * eps) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -4e-152:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (0.5 * eps) / x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -4e-152)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(0.5 * eps) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -4e-152)
		tmp = x - sqrt(-eps);
	else
		tmp = (0.5 * eps) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -4e-152], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * eps), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.00000000000000026e-152

   1. Initial program 98.6%

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

   3. Taylor expanded in eps around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -4.00000000000000026e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{0.5}{x} \cdot \varepsilon} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.3%

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

Alternative 5: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -4 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -4e-152)
   (- x (sqrt (- eps)))
   (* (/ 0.5 x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -4e-152) {
		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-152)) 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-152) {
		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-152:
		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-152)
		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-152)
		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-152], 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))) < -4.00000000000000026e-152

   1. Initial program 98.6%

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

   3. Taylor expanded in eps around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -4.00000000000000026e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 6: 56.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in eps around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 60.2

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

5. Applied rewrites 60.2%

   \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]
6. Add Preprocessing

Alternative 7: 3.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x - \left(-x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 3.4

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

5. Applied rewrites 3.4%

   \[\leadsto x - \color{blue}{\left(-x\right)} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024271 
(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))))