ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.9% → 99.0%

Time: 10.0s

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \frac{-0.125}{x \cdot x}, -0.5\right)}{x}, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-154)
     t_0
     (/ eps (+ x (fma eps (/ (fma eps (/ -0.125 (* x x)) -0.5) x) x))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + fma(eps, (fma(eps, (-0.125 / (x * x)), -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 <= -5e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + fma(eps, Float64(fma(eps, Float64(-0.125 / Float64(x * x)), -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, -5e-154], t$95$0, N[(eps / N[(x + N[(eps * N[(N[(eps * N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.0000000000000002e-154

   1. Initial program 99.3%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. --lowering--.f64 N/A

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

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

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

      13. --lowering--.f64 N/A

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

      14. *-lowering-*.f64 7.0

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

   4. Applied egg-rr 7.0%

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

   5. Taylor expanded in eps around 0

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

   7. Simplified 99.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \frac{-0.125}{x \cdot x}, -0.5\right)}{x}, x + x\right)}{\varepsilon}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 100.0

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

   9. Applied egg-rr 100.0%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{fma}\left(\varepsilon, \frac{-0.5}{x}, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-154) 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 <= -5e-154) {
		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 <= -5e-154)
		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, -5e-154], 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))) < -5.0000000000000002e-154

   1. Initial program 99.3%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. --lowering--.f64 N/A

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

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

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

      13. --lowering--.f64 N/A

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

      14. *-lowering-*.f64 7.0

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

   4. Applied egg-rr 7.0%

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

   5. Taylor expanded in eps around 0

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

   7. Simplified 99.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \frac{-0.125}{x \cdot x}, -0.5\right)}{x}, x + x\right)}{\varepsilon}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in eps around 0

       \[\leadsto \frac{\varepsilon}{x + \color{blue}{\left(x + \frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)}} \]
   11. 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. accelerator-lowering-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. /-lowering-/.f64 99.8

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

   12. Simplified 99.8%

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

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-154) t_0 (/ eps (+ x x)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + 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 <= (-5d-154)) then
        tmp = t_0
    else
        tmp = eps / (x + 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 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-154:
		tmp = t_0
	else:
		tmp = eps / (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 <= -5e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + 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 <= -5e-154)
		tmp = t_0;
	else
		tmp = eps / (x + 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, -5e-154], t$95$0, N[(eps / N[(x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.0000000000000002e-154

   1. Initial program 99.3%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. --lowering--.f64 N/A

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

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

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

      13. --lowering--.f64 N/A

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

      14. *-lowering-*.f64 7.0

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

   4. Applied egg-rr 7.0%

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

   5. Taylor expanded in eps around 0

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

   7. Simplified 99.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \frac{-0.125}{x \cdot x}, -0.5\right)}{x}, x + x\right)}{\varepsilon}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in eps around 0

       \[\leadsto \frac{\varepsilon}{x + \color{blue}{x}} \]
   11. Step-by-step derivation

   12. Simplified 98.7%

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

Alternative 4: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-154)
   (- x (sqrt (- eps)))
   (/ eps (+ x x))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-154) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + 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))) <= (-5d-154)) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + 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))) <= -5e-154) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-154:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-154)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-154)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + 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], -5e-154], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.0000000000000002e-154

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 96.5

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

   5. Simplified 96.5%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. --lowering--.f64 N/A

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

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

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

      13. --lowering--.f64 N/A

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

      14. *-lowering-*.f64 7.0

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

   4. Applied egg-rr 7.0%

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

   5. Taylor expanded in eps around 0

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

   7. Simplified 99.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \frac{-0.125}{x \cdot x}, -0.5\right)}{x}, x + x\right)}{\varepsilon}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in eps around 0

       \[\leadsto \frac{\varepsilon}{x + \color{blue}{x}} \]
   11. Step-by-step derivation

   12. Simplified 98.7%

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

Alternative 5: 44.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / (x + x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. /-lowering-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. clear-num N/A

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

   7. metadata-eval N/A

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

   8. flip-- N/A

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

   9. /-lowering-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. --lowering--.f64 N/A

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

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

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

   13. --lowering--.f64 N/A

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

   14. *-lowering-*.f64 60.3

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

4. Applied egg-rr 60.3%

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

5. Taylor expanded in eps around 0

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

7. Simplified 44.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \frac{-0.125}{x \cdot x}, -0.5\right)}{x}, x + x\right)}{\varepsilon}}} \]
8. Step-by-step derivation

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. *-lowering-*.f64 45.0

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

9. Applied egg-rr 45.0%

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

10. Taylor expanded in eps around 0

    \[\leadsto \frac{\varepsilon}{x + \color{blue}{x}} \]
11. Step-by-step derivation

12. Simplified 45.6%

    \[\leadsto \frac{\varepsilon}{x + \color{blue}{x}} \]
13. Add Preprocessing

Alternative 6: 7.9% accurate, 22.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

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

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

   4. /-lowering-/.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. metadata-eval N/A

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

   8. *-lowering-*.f64 45.6

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

5. Simplified 45.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f64 45.5

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

7. Applied egg-rr 45.5%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. unpow-prod-down N/A

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

   6. count-2 N/A

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

   7. flip-+ N/A

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

   8. difference-of-squares N/A

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

   9. +-inverses N/A

      \[\leadsto \varepsilon \cdot {\left(\frac{\left(x + x\right) \cdot \color{blue}{0}}{x - x}\right)}^{-1} \]

   10. +-inverses N/A

       \[\leadsto \varepsilon \cdot {\left(\frac{\left(x + x\right) \cdot 0}{\color{blue}{0}}\right)}^{-1} \]

   11. associate-*r/ N/A

       \[\leadsto \varepsilon \cdot {\color{blue}{\left(\left(x + x\right) \cdot \frac{0}{0}\right)}}^{-1} \]

   12. +-inverses N/A

       \[\leadsto \varepsilon \cdot {\left(\left(x + x\right) \cdot \frac{\color{blue}{x \cdot x - x \cdot x}}{0}\right)}^{-1} \]

   13. +-inverses N/A

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

   14. flip-+ N/A

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

   15. unpow-prod-down N/A

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

   16. inv-pow N/A

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

   17. flip-+ N/A

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

   18. +-inverses N/A

       \[\leadsto \varepsilon \cdot \left({\left(x + x\right)}^{-1} \cdot \frac{1}{\frac{\color{blue}{0}}{x - x}}\right) \]

   19. +-inverses N/A

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

   20. +-inverses N/A

       \[\leadsto \varepsilon \cdot \left({\left(x + x\right)}^{-1} \cdot \frac{1}{\frac{x - x}{\color{blue}{0}}}\right) \]

   21. +-inverses N/A

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

   22. clear-num N/A

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

   23. flip-+ N/A

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

   24. pow-plus N/A

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

   25. metadata-eval N/A

       \[\leadsto \varepsilon \cdot {\left(x + x\right)}^{\color{blue}{0}} \]

   26. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \color{blue}{1} \]

9. Applied egg-rr 7.8%

   \[\leadsto \color{blue}{\varepsilon \cdot 1} \]

10. Final simplification 7.8%

    \[\leadsto \varepsilon \]
11. Add Preprocessing

Alternative 7: 4.3% accurate, 22.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in x around inf

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

5. Simplified 4.3%

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

   1. +-inverses 4.3

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

7. Applied egg-rr 4.3%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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))))