ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.2% → 99.5%

Time: 8.9s

Alternatives: 10

Speedup: 0.6×

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: 61.2% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{-\varepsilon}{\left(-x\right) - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-neg.f64 59.1

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

4. Applied rewrites 59.1%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.6

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

7. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lift-neg.f64 N/A

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

   5. lower-fma.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Final simplification 99.6%

    \[\leadsto \frac{-\varepsilon}{\left(-x\right) - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}} \]
11. Add Preprocessing

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, \frac{x}{\varepsilon}, -1\right) \cdot \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{\mathsf{fma}\left(\frac{0.5}{x}, \varepsilon, -2 \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (sqrt (* (fma x (/ x eps) -1.0) eps)))
   (/ (- eps) (fma (/ 0.5 x) eps (* -2.0 x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - sqrt((fma(x, (x / eps), -1.0) * eps));
	} else {
		tmp = -eps / fma((0.5 / x), eps, (-2.0 * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(x - sqrt(Float64(fma(x, Float64(x / eps), -1.0) * eps)));
	else
		tmp = Float64(Float64(-eps) / fma(Float64(0.5 / x), eps, Float64(-2.0 * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-154], N[(x - N[Sqrt[N[(N[(x * N[(x / eps), $MachinePrecision] + -1.0), $MachinePrecision] * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-eps) / N[(N[(0.5 / x), $MachinePrecision] * eps + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-neg.f64 6.7

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 100.0

          \[\leadsto \frac{-\varepsilon}{\mathsf{fma}\left(\frac{0.5}{x}, \varepsilon, \color{blue}{-2 \cdot x}\right)} \]

   10. Applied rewrites 100.0%

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

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, \frac{x}{\varepsilon}, -1\right) \cdot \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (sqrt (* (fma x (/ x eps) -1.0) eps)))
   (/ (* 0.5 eps) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - sqrt((fma(x, (x / eps), -1.0) * 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))) <= -2e-154)
		tmp = Float64(x - sqrt(Float64(fma(x, Float64(x / eps), -1.0) * eps)));
	else
		tmp = Float64(Float64(0.5 * eps) / x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-154], N[(x - N[Sqrt[N[(N[(x * N[(x / eps), $MachinePrecision] + -1.0), $MachinePrecision] * eps), $MachinePrecision]], $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))) < -1.9999999999999999e-154

   1. Initial program 99.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-neg.f64 6.7

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.6%

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

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (sqrt (fma x x (- eps))))
   (/ (* 0.5 eps) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - sqrt(fma(x, x, -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))) <= -2e-154)
		tmp = Float64(x - sqrt(fma(x, x, Float64(-eps))));
	else
		tmp = Float64(Float64(0.5 * eps) / x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-154], N[(x - N[Sqrt[N[(x * x + (-eps)), $MachinePrecision]], $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))) < -1.9999999999999999e-154

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-neg.f64 6.7

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.6%

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

Alternative 5: 98.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154) 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 <= -2e-154) {
		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 <= (-2d-154)) 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 <= -2e-154) {
		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 <= -2e-154:
		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 <= -2e-154)
		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 <= -2e-154)
		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, -2e-154], 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))) < -1.9999999999999999e-154

   1. Initial program 99.2%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-neg.f64 6.7

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.6%

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

Alternative 6: 96.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (sqrt (- eps)))
   (/ (* 0.5 eps) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		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))) <= (-2d-154)) 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))) <= -2e-154) {
		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))) <= -2e-154:
		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))) <= -2e-154)
		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))) <= -2e-154)
		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], -2e-154], 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))) < -1.9999999999999999e-154

   1. Initial program 99.2%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-neg.f64 6.7

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\varepsilon}{x}} \]
   6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.6%

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

Alternative 7: 96.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (sqrt (- eps)))
   (* (/ 0.5 x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		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))) <= (-2d-154)) 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))) <= -2e-154) {
		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))) <= -2e-154:
		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))) <= -2e-154)
		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))) <= -2e-154)
		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], -2e-154], 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))) < -1.9999999999999999e-154

   1. Initial program 99.2%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.6%

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

   3. Taylor expanded in x around inf

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

      1. *-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 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 8: 99.5% accurate, 0.6× speedup

Math

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

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(Float64(-eps) / Float64(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 59.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-neg.f64 59.1

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

4. Applied rewrites 59.1%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{-\varepsilon}{\left(-x\right) - \sqrt{x \cdot x - \varepsilon}} \]
9. Add Preprocessing

Alternative 9: 56.4% 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 59.4%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 54.6

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

5. Applied rewrites 54.6%

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

Alternative 10: 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 59.4%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 3.5

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

5. Applied rewrites 3.5%

   \[\leadsto x - \color{blue}{\left(-x\right)} \]
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 2024295 
(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))))