ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.8% → 98.6%

Time: 5.1s

Alternatives: 7

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: 61.8% 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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-152)
   (/ 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))) <= -2e-152) {
		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))) <= -2e-152) {
		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))) <= -2e-152:
		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))) <= -2e-152)
		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))) <= -2e-152)
		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], -2e-152], 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))) < -2.00000000000000013e-152

   1. Initial program 99.3%

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

      1. flip-- 99.2%

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

      2. div-inv 98.8%

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

      3. add-sqr-sqrt 98.6%

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

      4. sub-neg 98.6%

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

      5. add-sqr-sqrt 98.6%

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

      6. hypot-def 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.8%

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

      2. *-rgt-identity 98.8%

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

      3. associate--r- 99.4%

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

      4. +-inverses 99.4%

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

      5. +-lft-identity 99.4%

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

   5. Simplified 99.4%

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

   if -2.00000000000000013e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 7.4%

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

      4. sub-neg 7.4%

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

      5. add-sqr-sqrt 2.1%

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

      6. hypot-def 2.1%

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

   3. Applied egg-rr 2.1%

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

      1. associate-*r/ 2.1%

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

      2. *-rgt-identity 2.1%

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

      3. associate--r- 40.2%

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

      4. +-inverses 40.2%

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

      5. +-lft-identity 40.2%

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

   5. Simplified 40.2%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-*r/ 99.5%

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

      10. *-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. fma-udef 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-152) 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 <= -2e-152) {
		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 <= (-2d-152)) 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 <= -2e-152) {
		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 <= -2e-152:
		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 <= -2e-152)
		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 <= -2e-152)
		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, -2e-152], 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))) < -2.00000000000000013e-152

   1. Initial program 99.3%

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

   if -2.00000000000000013e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.4%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 7.4%

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

      4. sub-neg 7.4%

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

      5. add-sqr-sqrt 2.1%

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

      6. hypot-def 2.1%

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

   3. Applied egg-rr 2.1%

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

      1. associate-*r/ 2.1%

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

      2. *-rgt-identity 2.1%

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

      3. associate--r- 40.2%

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

      4. +-inverses 40.2%

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

      5. +-lft-identity 40.2%

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

   5. Simplified 40.2%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.5%

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

      8. metadata-eval 99.5%

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

      9. associate-*r/ 99.5%

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

      10. *-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. fma-udef 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \]

Alternative 3: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-101}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.1e-101)
   (- x (sqrt (- eps)))
   (/ eps (+ (* x 2.0) (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.1e-101) {
		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 <= 3.1d-101) 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 <= 3.1e-101) {
		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 <= 3.1e-101:
		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 <= 3.1e-101)
		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 <= 3.1e-101)
		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, 3.1e-101], 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 < 3.09999999999999973e-101

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 94.9%

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

      1. neg-mul-1 94.9%

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

   4. Simplified 94.9%

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

   if 3.09999999999999973e-101 < x

   1. Initial program 28.5%

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

      1. flip-- 28.4%

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

      2. div-inv 28.4%

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

      3. add-sqr-sqrt 28.4%

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

      4. sub-neg 28.4%

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

      5. add-sqr-sqrt 24.2%

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

      6. hypot-def 24.2%

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

   3. Applied egg-rr 24.2%

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

      1. associate-*r/ 24.2%

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

      2. *-rgt-identity 24.2%

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

      3. associate--r- 54.5%

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

      4. +-inverses 54.5%

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

      5. +-lft-identity 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 80.4%

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

      6. *-commutative 80.4%

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

      7. associate-*r* 80.4%

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

      8. metadata-eval 80.4%

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

      9. associate-*r/ 80.4%

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

      10. *-commutative 80.4%

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

   8. Simplified 80.4%

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

      1. fma-udef 80.4%

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

   10. Applied egg-rr 80.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-101}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + \frac{\varepsilon}{x} \cdot -0.5}\\ \end{array} \]

Alternative 4: 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 64.5%

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

   1. flip-- 64.4%

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

   2. div-inv 64.2%

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

   3. add-sqr-sqrt 64.1%

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

   4. sub-neg 64.1%

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

   5. add-sqr-sqrt 62.1%

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

   6. hypot-def 62.1%

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

3. Applied egg-rr 62.1%

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

   1. associate-*r/ 62.1%

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

   2. *-rgt-identity 62.1%

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

   3. associate--r- 77.0%

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

   4. +-inverses 77.0%

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

   5. +-lft-identity 77.0%

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

5. Simplified 77.0%

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

6. Taylor expanded in x around inf 0.0%

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

   1. *-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. associate-*r/ 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 42.7%

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

   6. *-commutative 42.7%

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

   7. associate-*r* 42.7%

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

   8. metadata-eval 42.7%

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

   9. associate-*r/ 42.7%

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

   10. *-commutative 42.7%

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

8. Simplified 42.7%

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

   1. fma-udef 42.7%

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

10. Applied egg-rr 42.7%

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

11. Final simplification 42.7%

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

Alternative 5: 44.6% 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 64.5%

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

2. Taylor expanded in x around inf 41.9%

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

3. Final simplification 41.9%

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

Alternative 6: 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 64.5%

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

   1. flip-- 64.4%

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

   2. div-inv 64.2%

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

   3. add-sqr-sqrt 64.1%

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

   4. sub-neg 64.1%

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

   5. add-sqr-sqrt 62.1%

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

   6. hypot-def 62.1%

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

3. Applied egg-rr 62.1%

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

   1. associate-*r/ 62.1%

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

   2. *-rgt-identity 62.1%

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

   3. associate--r- 77.0%

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

   4. +-inverses 77.0%

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

   5. +-lft-identity 77.0%

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

5. Simplified 77.0%

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

6. Taylor expanded in x around inf 0.0%

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

   1. *-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. associate-*r/ 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 42.7%

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

   6. *-commutative 42.7%

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

   7. associate-*r* 42.7%

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

   8. metadata-eval 42.7%

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

   9. associate-*r/ 42.7%

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

   10. *-commutative 42.7%

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

8. Simplified 42.7%

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

9. Taylor expanded in eps around inf 5.4%

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

    1. *-commutative 5.4%

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

11. Simplified 5.4%

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

12. Final simplification 5.4%

    \[\leadsto x \cdot -2 \]

Alternative 7: 3.5% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x

Julia

function code(x, eps)
	return x
end

MATLAB

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

Wolfram

code[x_, eps_] := x

Derivation

1. Initial program 64.5%

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

2. Taylor expanded in x around 0 60.1%

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

   1. neg-mul-1 60.1%

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

4. Simplified 60.1%

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

5. Taylor expanded in x around inf 3.5%

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

6. Final simplification 3.5%

   \[\leadsto x \]

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 2023224 
(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))))