ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.3% → 98.5%

Time: 8.9s

Alternatives: 9

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: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, -0.125, -0.5\right)}{x}, \varepsilon, 2 \cdot x\right)}{\varepsilon}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153)
     t_0
     (pow
      (/ (fma (/ (fma (/ eps (* x x)) -0.125 -0.5) x) eps (* 2.0 x)) eps)
      -1.0))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = pow((fma((fma((eps / (x * x)), -0.125, -0.5) / x), eps, (2.0 * x)) / eps), -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-153)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(fma(Float64(eps / Float64(x * x)), -0.125, -0.5) / x), eps, Float64(2.0 * x)) / eps) ^ -1.0;
	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-153], t$95$0, N[Power[N[(N[(N[(N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.125 + -0.5), $MachinePrecision] / x), $MachinePrecision] * eps + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 97.1%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 6.9

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

   4. Applied rewrites 6.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 99.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{\varepsilon}{x}}{x}, -0.125, -0.5\right)}{x}, \varepsilon, 2 \cdot x\right)}{\varepsilon}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.5%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, -0.125, -0.5\right)}{x}, \varepsilon, 2 \cdot x\right)}{\varepsilon}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\varepsilon}{x \cdot x}, -0.125, -0.5\right)}{x}, \varepsilon, 2 \cdot x\right)}{\varepsilon}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(2, x, \frac{\varepsilon}{x} \cdot -0.5\right)}{\varepsilon}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153)
     t_0
     (pow (/ (fma 2.0 x (* (/ eps x) -0.5)) eps) -1.0))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = pow((fma(2.0, x, ((eps / x) * -0.5)) / eps), -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-153)
		tmp = t_0;
	else
		tmp = Float64(fma(2.0, x, Float64(Float64(eps / x) * -0.5)) / eps) ^ -1.0;
	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-153], t$95$0, N[Power[N[(N[(2.0 * x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 97.1%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 6.9

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

   4. Applied rewrites 6.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.5

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, x, \color{blue}{\frac{\varepsilon}{x}} \cdot -0.5\right)}{\varepsilon}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(2, x, \frac{\varepsilon}{x} \cdot -0.5\right)}{\varepsilon}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(2, x, \frac{\varepsilon}{x} \cdot -0.5\right)}{\varepsilon}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 2, \frac{-0.5}{x}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153) t_0 (pow (fma (/ x eps) 2.0 (/ -0.5 x)) -1.0))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = pow(fma((x / eps), 2.0, (-0.5 / x)), -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-153)
		tmp = t_0;
	else
		tmp = fma(Float64(x / eps), 2.0, Float64(-0.5 / x)) ^ -1.0;
	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-153], t$95$0, N[Power[N[(N[(x / eps), $MachinePrecision] * 2.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 97.1%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 6.9

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

   4. Applied rewrites 6.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.5

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, x, \color{blue}{\frac{\varepsilon}{x}} \cdot -0.5\right)}{\varepsilon}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(2, x, \frac{\varepsilon}{x} \cdot -0.5\right)}{\varepsilon}}} \]

   8. Taylor expanded in eps around inf

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

   10. Applied rewrites 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\varepsilon}, \color{blue}{2}, \frac{-0.5}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, 2, \frac{-0.5}{x}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{2}{\varepsilon}, x, \frac{-0.5}{x}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153) t_0 (pow (fma (/ 2.0 eps) x (/ -0.5 x)) -1.0))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = pow(fma((2.0 / eps), x, (-0.5 / x)), -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-153)
		tmp = t_0;
	else
		tmp = fma(Float64(2.0 / eps), x, Float64(-0.5 / x)) ^ -1.0;
	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-153], t$95$0, N[Power[N[(N[(2.0 / eps), $MachinePrecision] * x + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 97.1%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 6.9

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

   4. Applied rewrites 6.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.5

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, x, \color{blue}{\frac{\varepsilon}{x}} \cdot -0.5\right)}{\varepsilon}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(2, x, \frac{\varepsilon}{x} \cdot -0.5\right)}{\varepsilon}}} \]

   8. Taylor expanded in eps around inf

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

   10. Applied rewrites 99.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\varepsilon}, \color{blue}{2}, \frac{-0.5}{x}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{2}{\varepsilon}, x, \frac{-0.5}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{2}{\varepsilon}, x, \frac{-0.5}{x}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153) 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 <= -5e-153) {
		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 <= (-5d-153)) 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 <= -5e-153) {
		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 <= -5e-153:
		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 <= -5e-153)
		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 <= -5e-153)
		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, -5e-153], 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))) < -5.00000000000000033e-153

   1. Initial program 97.1%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 6.9

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

   4. Applied rewrites 6.9%

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

   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 98.4

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.9%

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

Alternative 6: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \varepsilon}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (- eps)))
   (/ (* 0.5 eps) x)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		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))) <= (-5d-153)) 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))) <= -5e-153) {
		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))) <= -5e-153:
		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))) <= -5e-153)
		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))) <= -5e-153)
		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], -5e-153], 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))) < -5.00000000000000033e-153

   1. Initial program 97.1%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 92.9

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

   5. Applied rewrites 92.9%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 6.9

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

   4. Applied rewrites 6.9%

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

   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 98.4

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.9%

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

Alternative 7: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -5e-153)
   (- x (sqrt (- eps)))
   (* (/ 0.5 x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-5d-153)) then
        tmp = x - sqrt(-eps)
    else
        tmp = (0.5d0 / x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -5e-153) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (0.5 / x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -5e-153:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (0.5 / x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -5e-153)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(0.5 / x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -5e-153)
		tmp = x - sqrt(-eps);
	else
		tmp = (0.5 / x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-153], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 97.1%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   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 92.9

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

   5. Applied rewrites 92.9%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 6.9%

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

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 56.3

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

5. Applied rewrites 56.3%

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

Alternative 9: 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 61.2%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
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 2024313 
(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))))