ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 99.3%

Time: 9.5s

Alternatives: 9

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: 61.7% 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: 99.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-154)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (+ (* x 2.0) (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -1e-154) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -1e-154:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-154)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(Float64(eps / x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	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-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[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 97.9%

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

      1. flip-- 97.9%

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

      2. div-inv 97.6%

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

      3. add-sqr-sqrt 97.2%

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

      4. associate--r- 99.1%

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

      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 99.1%

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

      9. hypot-define 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. +-inverses 99.2%

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

      3. +-lft-identity 99.2%

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

      4. associate-*l/ 99.1%

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

      5. *-lft-identity 99.1%

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

   6. Simplified 99.1%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.5%

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

      1. flip-- 8.5%

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

      2. div-inv 8.5%

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

      3. add-sqr-sqrt 8.6%

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

      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 47.5%

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

      9. hypot-define 47.5%

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

   4. Applied egg-rr 47.5%

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

      1. *-commutative 47.5%

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

      2. +-inverses 47.5%

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

      3. +-lft-identity 47.5%

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

      4. associate-*l/ 47.7%

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

      5. *-lft-identity 47.7%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in eps around 0 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. associate-*r/ 0.0%

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

      4. *-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.9%

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

      8. mul-1-neg 98.9%

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

      9. distribute-lft-neg-in 98.9%

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

      10. distribute-frac-neg 98.9%

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

      11. associate-*l/ 98.9%

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

      12. distribute-rgt-neg-in 98.9%

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

      13. metadata-eval 98.9%

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

   9. Simplified 98.9%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154) t_0 (/ eps (+ (* x 2.0) (* (/ eps x) -0.5))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-154) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	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-154)) then
        tmp = t_0
    else
        tmp = eps / ((x * 2.0d0) + ((eps / x) * (-0.5d0)))
    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-154) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-154:
		tmp = t_0
	else:
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(Float64(eps / x) * -0.5)));
	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-154)
		tmp = t_0;
	else
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	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-154], t$95$0, N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 97.9%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.5%

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

      1. flip-- 8.5%

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

      2. div-inv 8.5%

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

      3. add-sqr-sqrt 8.6%

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

      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 47.5%

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

      9. hypot-define 47.5%

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

   4. Applied egg-rr 47.5%

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

      1. *-commutative 47.5%

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

      2. +-inverses 47.5%

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

      3. +-lft-identity 47.5%

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

      4. associate-*l/ 47.7%

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

      5. *-lft-identity 47.7%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in eps around 0 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. associate-*r/ 0.0%

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

      4. *-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.9%

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

      8. mul-1-neg 98.9%

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

      9. distribute-lft-neg-in 98.9%

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

      10. distribute-frac-neg 98.9%

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

      11. associate-*l/ 98.9%

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

      12. distribute-rgt-neg-in 98.9%

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

      13. metadata-eval 98.9%

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

   9. Simplified 98.9%

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

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-115}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.1e-115)
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.1e-115) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.1d-115) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / ((x * 2.0d0) + ((eps / x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.1e-115) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.1e-115:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.1e-115)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(Float64(eps / x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.1e-115)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.1e-115], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000002e-115

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. neg-mul-1 97.7%

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

   5. Simplified 97.7%

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

   if 2.10000000000000002e-115 < x

   1. Initial program 25.2%

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

      1. flip-- 25.2%

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

      2. div-inv 25.1%

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

      3. add-sqr-sqrt 25.1%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 58.4%

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

      9. hypot-define 58.5%

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

   4. Applied egg-rr 58.5%

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

      1. *-commutative 58.5%

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

      2. +-inverses 58.5%

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

      3. +-lft-identity 58.5%

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

      4. associate-*l/ 58.7%

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

      5. *-lft-identity 58.7%

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

   6. Simplified 58.7%

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

   7. Taylor expanded in eps around 0 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. associate-*r/ 0.0%

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

      4. *-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 84.2%

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

      8. mul-1-neg 84.2%

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

      9. distribute-lft-neg-in 84.2%

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

      10. distribute-frac-neg 84.2%

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

      11. associate-*l/ 84.2%

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

      12. distribute-rgt-neg-in 84.2%

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

      13. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-117}:\\ \;\;\;\;-\sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 6e-117)
   (- (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 6e-117) {
		tmp = -sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 6d-117) then
        tmp = -sqrt(-eps)
    else
        tmp = eps / ((x * 2.0d0) + ((eps / x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 6e-117) {
		tmp = -Math.sqrt(-eps);
	} else {
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 6e-117:
		tmp = -math.sqrt(-eps)
	else:
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 6e-117)
		tmp = Float64(-sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(Float64(eps / x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 6e-117)
		tmp = -sqrt(-eps);
	else
		tmp = eps / ((x * 2.0) + ((eps / x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 6e-117], (-N[Sqrt[(-eps)], $MachinePrecision]), N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999982e-117

   1. Initial program 99.2%

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

      1. flip-- 99.1%

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

      2. div-inv 98.9%

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

      3. add-sqr-sqrt 98.4%

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

      4. associate--r- 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. sub-neg 99.2%

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

      8. add-sqr-sqrt 98.3%

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

      9. hypot-define 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. +-inverses 98.3%

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

      3. +-lft-identity 98.3%

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

      4. associate-*l/ 98.3%

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

      5. *-lft-identity 98.3%

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

   6. Simplified 98.3%

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

      1. add-cube-cbrt 97.6%

         \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \sqrt[3]{\varepsilon}}}\right)} \]

      2. distribute-rgt-neg-in 97.6%

         \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

      3. pow2 97.6%

         \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2}} \cdot \left(-\sqrt[3]{\varepsilon}\right)}\right)} \]

   8. Applied egg-rr 97.6%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2} \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

   9. Taylor expanded in eps around -inf 97.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{\varepsilon}{{\left(\sqrt[3]{-1}\right)}^{3}}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 97.2%

          \[\leadsto \color{blue}{-\sqrt{\frac{\varepsilon}{{\left(\sqrt[3]{-1}\right)}^{3}}}} \]

       2. rem-cube-cbrt 97.2%

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

       3. metadata-eval 97.2%

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

       4. distribute-neg-frac2 97.2%

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

       5. /-rgt-identity 97.2%

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

   11. Simplified 97.2%

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

   if 5.99999999999999982e-117 < x

   1. Initial program 25.2%

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

      1. flip-- 25.2%

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

      2. div-inv 25.1%

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

      3. add-sqr-sqrt 25.1%

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

      4. associate--r- 99.5%

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

      5. pow2 99.5%

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

      6. pow2 99.5%

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

      7. sub-neg 99.5%

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

      8. add-sqr-sqrt 58.4%

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

      9. hypot-define 58.5%

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

   4. Applied egg-rr 58.5%

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

      1. *-commutative 58.5%

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

      2. +-inverses 58.5%

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

      3. +-lft-identity 58.5%

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

      4. associate-*l/ 58.7%

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

      5. *-lft-identity 58.7%

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

   6. Simplified 58.7%

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

   7. Taylor expanded in eps around 0 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. associate-*r/ 0.0%

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

      4. *-commutative 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 84.2%

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

      8. mul-1-neg 84.2%

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

      9. distribute-lft-neg-in 84.2%

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

      10. distribute-frac-neg 84.2%

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

      11. associate-*l/ 84.2%

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

      12. distribute-rgt-neg-in 84.2%

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

      13. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

Alternative 5: 45.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

   1. flip-- 59.8%

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

   2. div-inv 59.7%

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

   3. add-sqr-sqrt 59.5%

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

   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 77.2%

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

   9. hypot-define 77.2%

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

4. Applied egg-rr 77.2%

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

   1. *-commutative 77.2%

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

   2. +-inverses 77.2%

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

   3. +-lft-identity 77.2%

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

   4. associate-*l/ 77.2%

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

   5. *-lft-identity 77.2%

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

6. Simplified 77.2%

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

7. Taylor expanded in eps around 0 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. associate-*r/ 0.0%

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

   4. *-commutative 0.0%

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

   5. *-commutative 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 47.8%

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

   8. mul-1-neg 47.8%

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

   9. distribute-lft-neg-in 47.8%

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

   10. distribute-frac-neg 47.8%

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

   11. associate-*l/ 47.8%

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

   12. distribute-rgt-neg-in 47.8%

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

   13. metadata-eval 47.8%

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

9. Simplified 47.8%

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

Alternative 6: 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 59.9%

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

   1. flip-- 59.8%

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

   2. div-inv 59.7%

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

   3. add-sqr-sqrt 59.5%

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

   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 77.2%

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

   9. hypot-define 77.2%

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

4. Applied egg-rr 77.2%

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

   1. *-commutative 77.2%

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

   2. +-inverses 77.2%

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

   3. +-lft-identity 77.2%

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

   4. associate-*l/ 77.2%

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

   5. *-lft-identity 77.2%

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

6. Simplified 77.2%

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

   1. add-cube-cbrt 76.8%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \sqrt[3]{\varepsilon}}}\right)} \]

   2. distribute-rgt-neg-in 76.8%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

   3. pow2 76.8%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2}} \cdot \left(-\sqrt[3]{\varepsilon}\right)}\right)} \]

8. Applied egg-rr 76.8%

   \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2} \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

9. Taylor expanded in eps around 0 0.0%

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

    1. associate-*r/ 0.0%

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

    2. remove-double-neg 0.0%

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

    3. neg-mul-1 0.0%

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

    4. times-frac 0.0%

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

    5. *-commutative 0.0%

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

    6. unpow2 0.0%

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

    7. rem-square-sqrt 47.7%

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

    8. neg-mul-1 47.7%

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

    9. distribute-neg-frac 47.7%

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

    10. distribute-frac-neg2 47.7%

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

    11. remove-double-neg 47.7%

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

    12. times-frac 47.7%

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

    13. *-commutative 47.7%

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

    14. neg-mul-1 47.7%

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

    15. associate-/l* 47.7%

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

    16. distribute-frac-neg2 47.7%

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

    17. distribute-neg-frac 47.7%

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

    18. metadata-eval 47.7%

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

11. Simplified 47.7%

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

Alternative 7: 44.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in x around inf 46.6%

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

4. Final simplification 46.6%

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

Alternative 8: 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 59.9%

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

   1. flip-- 59.8%

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

   2. div-inv 59.7%

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

   3. add-sqr-sqrt 59.5%

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

   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 77.2%

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

   9. hypot-define 77.2%

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

4. Applied egg-rr 77.2%

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

   1. *-commutative 77.2%

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

   2. +-inverses 77.2%

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

   3. +-lft-identity 77.2%

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

   4. associate-*l/ 77.2%

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

   5. *-lft-identity 77.2%

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

6. Simplified 77.2%

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

   1. add-cube-cbrt 76.8%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \sqrt[3]{\varepsilon}}}\right)} \]

   2. distribute-rgt-neg-in 76.8%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

   3. pow2 76.8%

      \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2}} \cdot \left(-\sqrt[3]{\varepsilon}\right)}\right)} \]

8. Applied egg-rr 76.8%

   \[\leadsto \frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{2} \cdot \left(-\sqrt[3]{\varepsilon}\right)}}\right)} \]

9. Taylor expanded in eps around 0 0.0%

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

    1. associate-*r/ 0.0%

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

    2. remove-double-neg 0.0%

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

    3. neg-mul-1 0.0%

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

    4. times-frac 0.0%

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

    5. *-commutative 0.0%

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

    6. unpow2 0.0%

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

    7. rem-square-sqrt 47.7%

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

    8. neg-mul-1 47.7%

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

    9. distribute-neg-frac 47.7%

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

    10. distribute-frac-neg2 47.7%

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

    11. remove-double-neg 47.7%

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

    12. times-frac 47.7%

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

    13. *-commutative 47.7%

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

    14. neg-mul-1 47.7%

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

    15. associate-/l* 47.7%

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

    16. distribute-frac-neg2 47.7%

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

    17. distribute-neg-frac 47.7%

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

    18. metadata-eval 47.7%

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

11. Simplified 47.7%

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

12. Taylor expanded in eps around inf 5.3%

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

    1. *-commutative 5.3%

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

14. Simplified 5.3%

    \[\leadsto \color{blue}{x \cdot -2} \]
15. Add Preprocessing

Alternative 9: 4.3% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in x around inf 4.3%

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

4. Taylor expanded in x around 0 4.3%

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

Developer Target 1: 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 2024136 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4d"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))

  :alt
  (! :herbie-platform default (/ eps (+ x (sqrt (- (* x x) eps)))))

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