ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.5% → 99.1%

Time: 9.4s

Alternatives: 11

Speedup: 1.0×

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: 61.5% 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: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-154:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.2%

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

      1. flip-- 96.1%

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

      2. div-inv 95.8%

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

      3. add-sqr-sqrt 95.8%

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

      4. associate--r- 99.4%

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

      5. pow2 99.4%

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

      6. pow2 99.4%

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

      7. sub-neg 99.4%

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

      8. add-sqr-sqrt 99.4%

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

      9. hypot-define 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.3%

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

      2. +-inverses 99.3%

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

      3. +-lft-identity 99.3%

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

      4. *-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

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

   1. Initial program 6.0%

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

      1. flip-- 5.9%

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

      2. div-inv 5.9%

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

      3. add-sqr-sqrt 6.0%

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

      4. associate--r- 99.4%

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

      5. pow2 99.4%

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

      6. pow2 99.4%

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

      7. sub-neg 99.4%

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

      8. add-sqr-sqrt 42.8%

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

      9. hypot-define 42.8%

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

   4. Applied egg-rr 42.8%

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

      1. associate-*r/ 43.0%

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

      2. +-inverses 43.0%

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

      3. +-lft-identity 43.0%

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

      4. *-rgt-identity 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} \cdot 0.5}} \]

      4. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon}}{x} \cdot 0.5} \]

      5. unpow2 0.0%

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

      6. rem-square-sqrt 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-1} \cdot \varepsilon}{x} \cdot 0.5} \]

      7. neg-mul-1 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon}}{x} \cdot 0.5} \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{-\varepsilon}{x} \cdot \color{blue}{\frac{-0.5}{-1}}} \]

      9. times-frac 100.0%

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

      10. distribute-lft-neg-in 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon \cdot -0.5}}{x \cdot -1}} \]

      11. distribute-rgt-neg-in 100.0%

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

      12. metadata-eval 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot \color{blue}{0.5}}{x \cdot -1}} \]

      13. times-frac 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon}{x} \cdot \frac{0.5}{-1}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot \color{blue}{-0.5}} \]

      15. associate-*l/ 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot -0.5}{x}}} \]

      16. associate-/l* 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\varepsilon \cdot \frac{-0.5}{x}}} \]

   9. Simplified 100.0%

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

4. Final simplification 99.6%

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

Alternative 2: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - Math.hypot(Math.sqrt(-eps), x);
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-154:
		tmp = x - math.hypot(math.sqrt(-eps), x)
	else:
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154)
		tmp = x - hypot(sqrt(-eps), x);
	else
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.2%

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

      1. sub-neg 96.2%

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

      2. +-commutative 96.2%

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

      3. add-sqr-sqrt 96.2%

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

      4. hypot-define 96.2%

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

   4. Applied egg-rr 96.2%

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

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

   1. Initial program 6.0%

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

      1. flip-- 5.9%

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

      2. div-inv 5.9%

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

      3. add-sqr-sqrt 6.0%

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

      4. associate--r- 99.4%

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

      5. pow2 99.4%

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

      6. pow2 99.4%

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

      7. sub-neg 99.4%

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

      8. add-sqr-sqrt 42.8%

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

      9. hypot-define 42.8%

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

   4. Applied egg-rr 42.8%

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

      1. associate-*r/ 43.0%

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

      2. +-inverses 43.0%

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

      3. +-lft-identity 43.0%

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

      4. *-rgt-identity 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} \cdot 0.5}} \]

      4. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon}}{x} \cdot 0.5} \]

      5. unpow2 0.0%

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

      6. rem-square-sqrt 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-1} \cdot \varepsilon}{x} \cdot 0.5} \]

      7. neg-mul-1 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon}}{x} \cdot 0.5} \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{-\varepsilon}{x} \cdot \color{blue}{\frac{-0.5}{-1}}} \]

      9. times-frac 100.0%

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

      10. distribute-lft-neg-in 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon \cdot -0.5}}{x \cdot -1}} \]

      11. distribute-rgt-neg-in 100.0%

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

      12. metadata-eval 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot \color{blue}{0.5}}{x \cdot -1}} \]

      13. times-frac 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon}{x} \cdot \frac{0.5}{-1}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot \color{blue}{-0.5}} \]

      15. associate-*l/ 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot -0.5}{x}}} \]

      16. associate-/l* 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\varepsilon \cdot \frac{-0.5}{x}}} \]

   9. Simplified 100.0%

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

4. Final simplification 97.8%

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

Alternative 3: 98.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154) t_0 (/ eps (+ (* x 2.0) (* eps (/ -0.5 x)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-2d-154)) then
        tmp = t_0
    else
        tmp = eps / ((x * 2.0d0) + (eps * ((-0.5d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-154:
		tmp = t_0
	else:
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -2e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(eps * Float64(-0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -2e-154)
		tmp = t_0;
	else
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-154], t$95$0, N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.2%

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

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

   1. Initial program 6.0%

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

      1. flip-- 5.9%

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

      2. div-inv 5.9%

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

      3. add-sqr-sqrt 6.0%

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

      4. associate--r- 99.4%

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

      5. pow2 99.4%

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

      6. pow2 99.4%

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

      7. sub-neg 99.4%

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

      8. add-sqr-sqrt 42.8%

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

      9. hypot-define 42.8%

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

   4. Applied egg-rr 42.8%

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

      1. associate-*r/ 43.0%

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

      2. +-inverses 43.0%

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

      3. +-lft-identity 43.0%

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

      4. *-rgt-identity 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} \cdot 0.5}} \]

      4. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon}}{x} \cdot 0.5} \]

      5. unpow2 0.0%

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

      6. rem-square-sqrt 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-1} \cdot \varepsilon}{x} \cdot 0.5} \]

      7. neg-mul-1 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon}}{x} \cdot 0.5} \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{-\varepsilon}{x} \cdot \color{blue}{\frac{-0.5}{-1}}} \]

      9. times-frac 100.0%

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

      10. distribute-lft-neg-in 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon \cdot -0.5}}{x \cdot -1}} \]

      11. distribute-rgt-neg-in 100.0%

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

      12. metadata-eval 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot \color{blue}{0.5}}{x \cdot -1}} \]

      13. times-frac 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon}{x} \cdot \frac{0.5}{-1}}} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot \color{blue}{-0.5}} \]

      15. associate-*l/ 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot -0.5}{x}}} \]

      16. associate-/l* 100.0%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\varepsilon \cdot \frac{-0.5}{x}}} \]

   9. Simplified 100.0%

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

4. Final simplification 97.8%

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

Alternative 4: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 4.3e-108)
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* eps (/ -0.5 x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 4.3e-108) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / 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 <= 4.3d-108) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / ((x * 2.0d0) + (eps * ((-0.5d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 4.3e-108) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 4.3e-108:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 4.3e-108)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(eps * Float64(-0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4.3e-108)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 4.3e-108], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.3e-108

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0 95.5%

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

      1. neg-mul-1 95.5%

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

   5. Simplified 95.5%

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

   if 4.3e-108 < x

   1. Initial program 18.4%

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

      1. flip-- 18.4%

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

      2. div-inv 18.3%

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

      3. add-sqr-sqrt 18.4%

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

      4. associate--r- 99.3%

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

      5. pow2 99.3%

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

      6. pow2 99.3%

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

      7. sub-neg 99.3%

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

      8. add-sqr-sqrt 53.6%

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

      9. hypot-define 53.6%

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

   4. Applied egg-rr 53.6%

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

      1. associate-*r/ 53.9%

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

      2. +-inverses 53.9%

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

      3. +-lft-identity 53.9%

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

      4. *-rgt-identity 53.9%

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

   6. Simplified 53.9%

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

   7. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

      3. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} \cdot 0.5}} \]

      4. *-commutative 0.0%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon}}{x} \cdot 0.5} \]

      5. unpow2 0.0%

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

      6. rem-square-sqrt 89.3%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-1} \cdot \varepsilon}{x} \cdot 0.5} \]

      7. neg-mul-1 89.3%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon}}{x} \cdot 0.5} \]

      8. metadata-eval 89.3%

         \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{-\varepsilon}{x} \cdot \color{blue}{\frac{-0.5}{-1}}} \]

      9. times-frac 89.3%

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

      10. distribute-lft-neg-in 89.3%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon \cdot -0.5}}{x \cdot -1}} \]

      11. distribute-rgt-neg-in 89.3%

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

      12. metadata-eval 89.3%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot \color{blue}{0.5}}{x \cdot -1}} \]

      13. times-frac 89.3%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon}{x} \cdot \frac{0.5}{-1}}} \]

      14. metadata-eval 89.3%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot \color{blue}{-0.5}} \]

      15. associate-*l/ 89.3%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot -0.5}{x}}} \]

      16. associate-/l* 89.3%

          \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\varepsilon \cdot \frac{-0.5}{x}}} \]

   9. Simplified 89.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-0.5}{x} + 2 \cdot \frac{x}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ 1.0 (+ (/ -0.5 x) (* 2.0 (/ x eps)))))

C

double code(double x, double eps) {
	return 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 / (((-0.5d0) / x) + (2.0d0 * (x / eps)))
end function

Java

public static double code(double x, double eps) {
	return 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
}

Python

def code(x, eps):
	return 1.0 / ((-0.5 / x) + (2.0 * (x / eps)))

Julia

function code(x, eps)
	return Float64(1.0 / Float64(Float64(-0.5 / x) + Float64(2.0 * Float64(x / eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
end

Wolfram

code[x_, eps_] := N[(1.0 / N[(N[(-0.5 / x), $MachinePrecision] + N[(2.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

   1. flip-- 58.4%

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

   2. clear-num 58.2%

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

   3. sub-neg 58.2%

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

   4. add-sqr-sqrt 56.6%

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

   5. hypot-define 56.6%

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

   6. add-sqr-sqrt 56.6%

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

   7. associate--r- 75.7%

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

   8. pow2 75.7%

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

   9. pow2 75.7%

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

4. Applied egg-rr 75.7%

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

5. Taylor expanded in x around inf 0.0%

   \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot \frac{x}{\varepsilon}}} \]
6. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} + 0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   2. associate-*r/ 0.0%

      \[\leadsto \frac{1}{2 \cdot \frac{x}{\varepsilon} + \color{blue}{\frac{0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   3. unpow2 0.0%

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

   4. rem-square-sqrt 48.4%

      \[\leadsto \frac{1}{2 \cdot \frac{x}{\varepsilon} + \frac{0.5 \cdot \color{blue}{-1}}{x}} \]

   5. metadata-eval 48.4%

      \[\leadsto \frac{1}{2 \cdot \frac{x}{\varepsilon} + \frac{\color{blue}{-0.5}}{x}} \]

7. Simplified 48.4%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} + \frac{-0.5}{x}}} \]

8. Final simplification 48.4%

   \[\leadsto \frac{1}{\frac{-0.5}{x} + 2 \cdot \frac{x}{\varepsilon}} \]
9. Add Preprocessing

Alternative 6: 45.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ (* x 2.0) (* eps (/ -0.5 x)))))

C

double code(double x, double eps) {
	return eps / ((x * 2.0) + (eps * (-0.5 / x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / ((x * 2.0d0) + (eps * ((-0.5d0) / x)))
end function

Java

public static double code(double x, double eps) {
	return eps / ((x * 2.0) + (eps * (-0.5 / x)));
}

Python

def code(x, eps):
	return eps / ((x * 2.0) + (eps * (-0.5 / x)))

Julia

function code(x, eps)
	return Float64(eps / Float64(Float64(x * 2.0) + Float64(eps * Float64(-0.5 / x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / ((x * 2.0) + (eps * (-0.5 / x)));
end

Wolfram

code[x_, eps_] := N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(eps * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

   1. flip-- 58.4%

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

   2. div-inv 58.3%

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

   3. add-sqr-sqrt 58.2%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 75.7%

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

   9. hypot-define 75.7%

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

4. Applied egg-rr 75.7%

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

   1. associate-*r/ 75.8%

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

   2. +-inverses 75.8%

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

   3. +-lft-identity 75.8%

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

   4. *-rgt-identity 75.8%

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

6. Simplified 75.8%

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

7. Taylor expanded in x around inf 0.0%

   \[\leadsto \frac{\varepsilon}{\color{blue}{0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot x}} \]
8. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   2. *-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2} + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}} \]

   3. *-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x} \cdot 0.5}} \]

   4. *-commutative 0.0%

      \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \varepsilon}}{x} \cdot 0.5} \]

   5. unpow2 0.0%

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

   6. rem-square-sqrt 48.6%

      \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-1} \cdot \varepsilon}{x} \cdot 0.5} \]

   7. neg-mul-1 48.6%

      \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon}}{x} \cdot 0.5} \]

   8. metadata-eval 48.6%

      \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{-\varepsilon}{x} \cdot \color{blue}{\frac{-0.5}{-1}}} \]

   9. times-frac 48.6%

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

   10. distribute-lft-neg-in 48.6%

       \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\color{blue}{-\varepsilon \cdot -0.5}}{x \cdot -1}} \]

   11. distribute-rgt-neg-in 48.6%

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

   12. metadata-eval 48.6%

       \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon \cdot \color{blue}{0.5}}{x \cdot -1}} \]

   13. times-frac 48.6%

       \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon}{x} \cdot \frac{0.5}{-1}}} \]

   14. metadata-eval 48.6%

       \[\leadsto \frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot \color{blue}{-0.5}} \]

   15. associate-*l/ 48.6%

       \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\frac{\varepsilon \cdot -0.5}{x}}} \]

   16. associate-/l* 48.6%

       \[\leadsto \frac{\varepsilon}{x \cdot 2 + \color{blue}{\varepsilon \cdot \frac{-0.5}{x}}} \]

9. Simplified 48.6%

   \[\leadsto \frac{\varepsilon}{\color{blue}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}}} \]

10. Final simplification 48.6%

    \[\leadsto \frac{\varepsilon}{x \cdot 2 + \varepsilon \cdot \frac{-0.5}{x}} \]
11. Add Preprocessing

Alternative 7: 44.7% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (/ 0.5 x)))

C

double code(double x, double eps) {
	return eps * (0.5 / x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (0.5d0 / x)
end function

Java

public static double code(double x, double eps) {
	return eps * (0.5 / x);
}

Python

def code(x, eps):
	return eps * (0.5 / x)

Julia

function code(x, eps)
	return Float64(eps * Float64(0.5 / x))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (0.5 / x);
end

Wolfram

code[x_, eps_] := N[(eps * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

   1. flip-- 58.4%

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

   2. clear-num 58.2%

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

   3. sub-neg 58.2%

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

   4. add-sqr-sqrt 56.6%

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

   5. hypot-define 56.6%

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

   6. add-sqr-sqrt 56.6%

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

   7. associate--r- 75.7%

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

   8. pow2 75.7%

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

   9. pow2 75.7%

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

4. Applied egg-rr 75.7%

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

5. Taylor expanded in x around inf 47.5%

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

   1. associate-*r/ 47.5%

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

   2. *-commutative 47.5%

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

7. Simplified 47.5%

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

   1. associate-/r/ 47.4%

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

   2. *-commutative 47.4%

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

   3. associate-/r* 47.4%

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

   4. metadata-eval 47.4%

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

9. Applied egg-rr 47.4%

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

10. Final simplification 47.4%

    \[\leadsto \varepsilon \cdot \frac{0.5}{x} \]
11. Add Preprocessing

Alternative 8: 44.6% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{x}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ 0.5 (/ x eps)))

C

double code(double x, double eps) {
	return 0.5 / (x / eps);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.5d0 / (x / eps)
end function

Java

public static double code(double x, double eps) {
	return 0.5 / (x / eps);
}

Python

def code(x, eps):
	return 0.5 / (x / eps)

Julia

function code(x, eps)
	return Float64(0.5 / Float64(x / eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 / (x / eps);
end

Wolfram

code[x_, eps_] := N[(0.5 / N[(x / eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

   1. flip-- 58.4%

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

   2. clear-num 58.2%

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

   3. sub-neg 58.2%

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

   4. add-sqr-sqrt 56.6%

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

   5. hypot-define 56.6%

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

   6. add-sqr-sqrt 56.6%

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

   7. associate--r- 75.7%

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

   8. pow2 75.7%

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

   9. pow2 75.7%

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

4. Applied egg-rr 75.7%

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

5. Taylor expanded in x around inf 47.5%

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

   1. associate-*r/ 47.5%

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

   2. *-commutative 47.5%

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

7. Simplified 47.5%

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

8. Taylor expanded in x around 0 47.7%

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

   1. *-lft-identity 47.7%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{1 \cdot \varepsilon}}{x} \]

   2. associate-*l/ 47.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{x} \cdot \varepsilon\right)} \]

   3. associate-/r/ 47.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{x}{\varepsilon}}} \]

   4. associate-*r/ 47.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{\frac{x}{\varepsilon}}} \]

   5. metadata-eval 47.5%

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

10. Simplified 47.5%

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

11. Final simplification 47.5%

    \[\leadsto \frac{0.5}{\frac{x}{\varepsilon}} \]
12. Add Preprocessing

Alternative 9: 44.8% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot 0.5}{x} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (* eps 0.5) x))

C

double code(double x, double eps) {
	return (eps * 0.5) / x;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * 0.5d0) / x
end function

Java

public static double code(double x, double eps) {
	return (eps * 0.5) / x;
}

Python

def code(x, eps):
	return (eps * 0.5) / x

Julia

function code(x, eps)
	return Float64(Float64(eps * 0.5) / x)
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * 0.5) / x;
end

Wolfram

code[x_, eps_] := N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in x around inf 47.7%

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

   1. associate-*r/ 47.7%

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

5. Simplified 47.7%

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

6. Final simplification 47.7%

   \[\leadsto \frac{\varepsilon \cdot 0.5}{x} \]
7. Add Preprocessing

Alternative 10: 5.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* x -2.0))

C

double code(double x, double eps) {
	return x * -2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return x * -2.0;
}

Python

def code(x, eps):
	return x * -2.0

Julia

function code(x, eps)
	return Float64(x * -2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = x * -2.0;
end

Wolfram

code[x_, eps_] := N[(x * -2.0), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

   1. flip-- 58.4%

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

   2. clear-num 58.2%

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

   3. sub-neg 58.2%

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

   4. add-sqr-sqrt 56.6%

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

   5. hypot-define 56.6%

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

   6. add-sqr-sqrt 56.6%

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

   7. associate--r- 75.7%

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

   8. pow2 75.7%

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

   9. pow2 75.7%

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

4. Applied egg-rr 75.7%

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

5. Taylor expanded in x around inf 0.0%

   \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot \frac{x}{\varepsilon}}} \]
6. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} + 0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   2. associate-*r/ 0.0%

      \[\leadsto \frac{1}{2 \cdot \frac{x}{\varepsilon} + \color{blue}{\frac{0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   3. unpow2 0.0%

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

   4. rem-square-sqrt 48.4%

      \[\leadsto \frac{1}{2 \cdot \frac{x}{\varepsilon} + \frac{0.5 \cdot \color{blue}{-1}}{x}} \]

   5. metadata-eval 48.4%

      \[\leadsto \frac{1}{2 \cdot \frac{x}{\varepsilon} + \frac{\color{blue}{-0.5}}{x}} \]

7. Simplified 48.4%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} + \frac{-0.5}{x}}} \]

8. Taylor expanded in x around 0 5.1%

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

   1. *-commutative 5.1%

      \[\leadsto \color{blue}{x \cdot -2} \]

10. Simplified 5.1%

    \[\leadsto \color{blue}{x \cdot -2} \]

11. Final simplification 5.1%

    \[\leadsto x \cdot -2 \]
12. Add Preprocessing

Alternative 11: 3.5% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x

Julia

function code(x, eps)
	return x
end

MATLAB

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

Wolfram

code[x_, eps_] := x

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in eps around inf 3.5%

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

4. Final simplification 3.5%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 99.5% accurate, 1.0× 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 2024046 
(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
  (/ eps (+ x (sqrt (- (* x x) eps))))

  (- x (sqrt (- (* x x) eps))))