ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.8% → 99.0%

Time: 7.2s

Alternatives: 7

Speedup: 0.5×

Specification

\[\left(0 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-153) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / (x + (x - (eps * (0.5 / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-153:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / (x + (x - (eps * (0.5 / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-153)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(x + Float64(x - 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))) <= -1e-153)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / (x + (x - (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], -1e-153], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x - N[(eps * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.7%

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

      1. flip-- 98.6%

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

      2. div-inv 98.3%

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

      3. add-sqr-sqrt 97.9%

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

      4. associate--r- 99.1%

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

      5. pow2 99.1%

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

      6. pow2 99.1%

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

      7. sub-neg 99.1%

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

      8. add-sqr-sqrt 99.2%

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

      9. hypot-def 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. +-inverses 99.1%

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

      2. +-lft-identity 99.1%

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

      3. associate-*r/ 99.2%

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

      4. *-rgt-identity 99.2%

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

   6. Simplified 99.2%

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

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

   1. Initial program 6.1%

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

      1. flip-- 6.1%

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

      2. div-inv 6.1%

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

      3. add-sqr-sqrt 6.2%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 53.9%

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

      9. hypot-def 53.9%

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

   4. Applied egg-rr 53.9%

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

      1. +-inverses 53.9%

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

      2. +-lft-identity 53.9%

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

      3. associate-*r/ 54.1%

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

      4. *-rgt-identity 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 100.0%

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

      6. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

       1. fma-udef 100.0%

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

       2. frac-2neg 100.0%

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

       3. remove-double-neg 100.0%

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

       4. associate-*r/ 100.0%

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

       5. metadata-eval 100.0%

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

       6. distribute-lft-neg-in 100.0%

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

       7. frac-2neg 100.0%

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

       8. +-commutative 100.0%

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

       9. add-sqr-sqrt 45.9%

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

       10. sqrt-unprod 99.7%

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

       11. sqr-neg 99.7%

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

       12. sqrt-unprod 54.1%

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

       13. add-sqr-sqrt 99.3%

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

       14. distribute-rgt-neg-in 99.3%

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

       15. distribute-lft-neg-in 99.3%

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

       16. metadata-eval 99.3%

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

       17. associate-*r/ 99.3%

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

       18. metadata-eval 99.3%

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

       19. cancel-sign-sub-inv 99.3%

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

       20. associate-*r/ 99.3%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.5%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-153:
		tmp = t_0
	else:
		tmp = eps / (x + (x - (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 <= -1e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x - 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 <= -1e-153)
		tmp = t_0;
	else
		tmp = eps / (x + (x - (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, -1e-153], t$95$0, N[(eps / N[(x + N[(x - N[(eps * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.7%

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

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

   1. Initial program 6.1%

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

      1. flip-- 6.1%

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

      2. div-inv 6.1%

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

      3. add-sqr-sqrt 6.2%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 53.9%

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

      9. hypot-def 53.9%

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

   4. Applied egg-rr 53.9%

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

      1. +-inverses 53.9%

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

      2. +-lft-identity 53.9%

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

      3. associate-*r/ 54.1%

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

      4. *-rgt-identity 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 100.0%

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

      6. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

       1. fma-udef 100.0%

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

       2. frac-2neg 100.0%

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

       3. remove-double-neg 100.0%

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

       4. associate-*r/ 100.0%

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

       5. metadata-eval 100.0%

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

       6. distribute-lft-neg-in 100.0%

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

       7. frac-2neg 100.0%

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

       8. +-commutative 100.0%

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

       9. add-sqr-sqrt 45.9%

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

       10. sqrt-unprod 99.7%

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

       11. sqr-neg 99.7%

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

       12. sqrt-unprod 54.1%

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

       13. add-sqr-sqrt 99.3%

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

       14. distribute-rgt-neg-in 99.3%

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

       15. distribute-lft-neg-in 99.3%

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

       16. metadata-eval 99.3%

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

       17. associate-*r/ 99.3%

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

       18. metadata-eval 99.3%

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

       19. cancel-sign-sub-inv 99.3%

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

       20. associate-*r/ 99.3%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

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

Alternative 3: 45.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

   1. flip-- 59.2%

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

   2. div-inv 59.0%

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

   3. add-sqr-sqrt 58.8%

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

   4. associate--r- 99.3%

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

   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 79.9%

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

   9. hypot-def 79.9%

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

4. Applied egg-rr 79.9%

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

   1. +-inverses 79.9%

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

   2. +-lft-identity 79.9%

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

   3. associate-*r/ 80.0%

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

   4. *-rgt-identity 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 47.0%

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

   6. neg-mul-1 47.0%

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

9. Simplified 47.0%

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

    1. fma-udef 47.0%

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

    2. frac-2neg 47.0%

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

    3. remove-double-neg 47.0%

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

    4. associate-*r/ 47.0%

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

    5. metadata-eval 47.0%

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

    6. distribute-lft-neg-in 47.0%

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

    7. frac-2neg 47.0%

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

    8. +-commutative 47.0%

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

    9. add-sqr-sqrt 19.5%

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

    10. sqrt-unprod 45.6%

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

    11. sqr-neg 45.6%

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

    12. sqrt-unprod 25.6%

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

    13. add-sqr-sqrt 44.9%

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

    14. distribute-rgt-neg-in 44.9%

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

    15. distribute-lft-neg-in 44.9%

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

    16. metadata-eval 44.9%

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

    17. associate-*r/ 44.9%

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

    18. metadata-eval 44.9%

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

    19. cancel-sign-sub-inv 44.9%

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

    20. associate-*r/ 44.9%

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

11. Applied egg-rr 47.0%

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

13. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{x \cdot 2}{\varepsilon} - \frac{0.5}{x}}\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 46.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{x \cdot 2}{\varepsilon} - \frac{0.5}{x}}\right)\right)} \]

    2. expm1-log1p 46.8%

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

    3. sub-neg 46.8%

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

    4. /-rgt-identity 46.8%

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

    5. associate-/l* 46.8%

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

    6. metadata-eval 46.8%

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

    7. associate-/r* 46.8%

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

    8. distribute-neg-frac 46.8%

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

    9. metadata-eval 46.8%

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

15. Simplified 46.8%

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

16. Final simplification 46.8%

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

Alternative 4: 45.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \left(x - \varepsilon \cdot \frac{0.5}{x}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

   1. flip-- 59.2%

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

   2. div-inv 59.0%

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

   3. add-sqr-sqrt 58.8%

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

   4. associate--r- 99.3%

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

   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 79.9%

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

   9. hypot-def 79.9%

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

4. Applied egg-rr 79.9%

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

   1. +-inverses 79.9%

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

   2. +-lft-identity 79.9%

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

   3. associate-*r/ 80.0%

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

   4. *-rgt-identity 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 47.0%

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

   6. neg-mul-1 47.0%

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

9. Simplified 47.0%

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

    1. fma-udef 47.0%

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

    2. frac-2neg 47.0%

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

    3. remove-double-neg 47.0%

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

    4. associate-*r/ 47.0%

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

    5. metadata-eval 47.0%

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

    6. distribute-lft-neg-in 47.0%

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

    7. frac-2neg 47.0%

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

    8. +-commutative 47.0%

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

    9. add-sqr-sqrt 19.5%

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

    10. sqrt-unprod 45.6%

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

    11. sqr-neg 45.6%

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

    12. sqrt-unprod 25.6%

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

    13. add-sqr-sqrt 44.9%

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

    14. distribute-rgt-neg-in 44.9%

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

    15. distribute-lft-neg-in 44.9%

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

    16. metadata-eval 44.9%

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

    17. associate-*r/ 44.9%

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

    18. metadata-eval 44.9%

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

    19. cancel-sign-sub-inv 44.9%

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

    20. associate-*r/ 44.9%

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

11. Applied egg-rr 47.0%

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

12. Final simplification 47.0%

    \[\leadsto \frac{\varepsilon}{x + \left(x - \varepsilon \cdot \frac{0.5}{x}\right)} \]
13. Add Preprocessing

Alternative 5: 44.3% 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 59.3%

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

   1. flip-- 59.2%

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

   2. div-inv 59.0%

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

   3. add-sqr-sqrt 58.8%

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

   4. associate--r- 99.3%

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

   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 79.9%

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

   9. hypot-def 79.9%

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

4. Applied egg-rr 79.9%

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

   1. +-inverses 79.9%

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

   2. +-lft-identity 79.9%

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

   3. associate-*r/ 80.0%

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

   4. *-rgt-identity 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in eps around 0 46.5%

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

   1. associate-*r/ 46.5%

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

   2. associate-/l* 46.3%

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

9. Simplified 46.3%

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

    1. associate-/r/ 46.3%

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

11. Applied egg-rr 46.3%

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

12. Final simplification 46.3%

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

Alternative 6: 44.5% 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 59.3%

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

3. Taylor expanded in x around inf 46.5%

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

   1. associate-*r/ 46.5%

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

5. Simplified 46.5%

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

6. Final simplification 46.5%

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

Alternative 7: 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 59.3%

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

   1. flip-- 59.2%

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

   2. div-inv 59.0%

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

   3. add-sqr-sqrt 58.8%

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

   4. associate--r- 99.3%

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

   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 79.9%

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

   9. hypot-def 79.9%

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

4. Applied egg-rr 79.9%

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

   1. +-inverses 79.9%

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

   2. +-lft-identity 79.9%

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

   3. associate-*r/ 80.0%

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

   4. *-rgt-identity 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in x around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 47.0%

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

   6. neg-mul-1 47.0%

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

9. Simplified 47.0%

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

10. Taylor expanded in eps around inf 5.2%

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

    1. *-commutative 5.2%

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

12. Simplified 5.2%

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

13. Final simplification 5.2%

    \[\leadsto x \cdot -2 \]
14. 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 2024019 
(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)))

  :herbie-target
  (/ eps (+ x (sqrt (- (* x x) eps))))

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