ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 99.5%

Time: 12.8s

Alternatives: 6

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: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

4. Applied rewrites 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. +-inverses N/A

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

   4. +-rgt-identity 99.5

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

8. Applied rewrites 99.5%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-153) {
		tmp = t_0;
	} else {
		tmp = eps / (x + fma(eps, (-0.5 / x), x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + fma(eps, Float64(-0.5 / x), x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-153], t$95$0, N[(eps / N[(x + N[(eps * N[(-0.5 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.6%

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

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

   1. Initial program 8.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 8.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. +-inverses N/A

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

      4. +-rgt-identity 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. metadata-eval N/A

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

       6. distribute-neg-frac N/A

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

       7. metadata-eval N/A

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

       8. associate-*r/ N/A

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

       9. lower-fma.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. distribute-neg-frac N/A

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

       13. metadata-eval N/A

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

       14. lower-/.f64 98.6

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

   11. Applied rewrites 98.6%

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

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 2024228 
(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))))