ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.2% → 98.2%

Time: 11.4s

Alternatives: 10

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: 62.2% 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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{x \cdot -2 + 0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-151) t_0 (/ (- eps) (+ (* x -2.0) (* 0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-151) {
		tmp = t_0;
	} else {
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-151)) then
        tmp = t_0
    else
        tmp = -eps / ((x * (-2.0d0)) + (0.5d0 * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-151) {
		tmp = t_0;
	} else {
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-151:
		tmp = t_0
	else:
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-151)
		tmp = t_0;
	else
		tmp = Float64(Float64(-eps) / Float64(Float64(x * -2.0) + Float64(0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-151)
		tmp = t_0;
	else
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-151], t$95$0, N[((-eps) / N[(N[(x * -2.0), $MachinePrecision] + N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999994e-152

   1. Initial program 99.9%

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

   if -9.9999999999999994e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.2%

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

      1. sub-neg 9.2%

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

      2. +-commutative 9.2%

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

      3. flip-+ 9.3%

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

      4. sqr-neg 9.3%

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

      5. rem-square-sqrt 9.3%

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

   3. Applied egg-rr 9.3%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 99.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{x \cdot -2 + 0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \]

Alternative 2: 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]

Derivation

1. Initial program 63.4%

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

   1. sub-neg 63.4%

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

   2. +-commutative 63.4%

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

   3. flip-+ 63.4%

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

   4. sqr-neg 63.4%

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

   5. rem-square-sqrt 63.1%

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

3. Applied egg-rr 63.1%

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

4. Taylor expanded in x around 0 99.6%

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

   1. mul-1-neg 99.6%

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

6. Simplified 99.6%

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

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{\varepsilon}{\sqrt{x \cdot x - \varepsilon} + x}} \]
9. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. +-commutative 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\varepsilon}{x + \sqrt{-\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{x \cdot -2 + 0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.2e-103)
   (/ eps (+ x (sqrt (- eps))))
   (/ (- eps) (+ (* x -2.0) (* 0.5 (/ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.2e-103) {
		tmp = eps / (x + sqrt(-eps));
	} else {
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.2d-103) then
        tmp = eps / (x + sqrt(-eps))
    else
        tmp = -eps / ((x * (-2.0d0)) + (0.5d0 * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.2e-103) {
		tmp = eps / (x + Math.sqrt(-eps));
	} else {
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.2e-103:
		tmp = eps / (x + math.sqrt(-eps))
	else:
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.2e-103)
		tmp = Float64(eps / Float64(x + sqrt(Float64(-eps))));
	else
		tmp = Float64(Float64(-eps) / Float64(Float64(x * -2.0) + Float64(0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.2e-103)
		tmp = eps / (x + sqrt(-eps));
	else
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.2e-103], N[(eps / N[(x + N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-eps) / N[(N[(x * -2.0), $MachinePrecision] + N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1999999999999999e-103

   1. Initial program 95.7%

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

      1. sub-neg 95.7%

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

      2. +-commutative 95.7%

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

      3. flip-+ 95.7%

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

      4. sqr-neg 95.7%

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

      5. rem-square-sqrt 95.1%

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

   3. Applied egg-rr 95.1%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. mul-1-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 92.6%

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

      1. mul-1-neg 5.5%

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

   9. Simplified 92.6%

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

       1. frac-2neg 92.6%

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

       2. remove-double-neg 92.6%

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

       3. div-inv 92.5%

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

   11. Applied egg-rr 92.5%

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

       1. associate-*r/ 92.6%

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

       2. *-rgt-identity 92.6%

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

       3. sub-neg 92.6%

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

       4. distribute-neg-in 92.6%

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

       5. remove-double-neg 92.6%

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

       6. +-commutative 92.6%

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

   13. Simplified 92.6%

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

   if 2.1999999999999999e-103 < x

   1. Initial program 19.9%

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

      1. sub-neg 19.9%

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

      2. +-commutative 19.9%

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

      3. flip-+ 19.9%

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

      4. sqr-neg 19.9%

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

      5. rem-square-sqrt 19.9%

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

   3. Applied egg-rr 19.9%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 87.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\varepsilon}{x + \sqrt{-\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{x \cdot -2 + 0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \]

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-103}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{x \cdot -2 + 0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 4.4e-103)
   (- x (sqrt (- eps)))
   (/ (- eps) (+ (* x -2.0) (* 0.5 (/ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 4.4e-103) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 4.4d-103) then
        tmp = x - sqrt(-eps)
    else
        tmp = -eps / ((x * (-2.0d0)) + (0.5d0 * (eps / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 4.4e-103) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 4.4e-103:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 4.4e-103)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(-eps) / Float64(Float64(x * -2.0) + Float64(0.5 * Float64(eps / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4.4e-103)
		tmp = x - sqrt(-eps);
	else
		tmp = -eps / ((x * -2.0) + (0.5 * (eps / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 4.4e-103], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[((-eps) / N[(N[(x * -2.0), $MachinePrecision] + N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.3999999999999999e-103

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0 92.6%

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

      1. mul-1-neg 5.5%

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

   4. Simplified 92.6%

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

   if 4.3999999999999999e-103 < x

   1. Initial program 19.9%

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

      1. sub-neg 19.9%

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

      2. +-commutative 19.9%

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

      3. flip-+ 19.9%

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

      4. sqr-neg 19.9%

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

      5. rem-square-sqrt 19.9%

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

   3. Applied egg-rr 19.9%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 87.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-103}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{x \cdot -2 + 0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \]

Alternative 5: 44.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. sub-neg 63.4%

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

   2. +-commutative 63.4%

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

   3. flip-+ 63.4%

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

   4. sqr-neg 63.4%

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

   5. rem-square-sqrt 63.1%

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

3. Applied egg-rr 63.1%

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

4. Taylor expanded in x around 0 99.6%

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

   1. mul-1-neg 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around inf 43.6%

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

8. Final simplification 43.6%

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

Alternative 6: 44.9% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. sub-neg 63.4%

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

   2. +-commutative 63.4%

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

   3. flip-+ 63.4%

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

   4. sqr-neg 63.4%

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

   5. rem-square-sqrt 63.1%

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

3. Applied egg-rr 63.1%

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

4. Taylor expanded in x around 0 99.6%

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

   1. mul-1-neg 99.6%

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

6. Simplified 99.6%

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

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{\varepsilon}{\sqrt{x \cdot x - \varepsilon} + x}} \]
9. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. +-commutative 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in x around inf 43.6%

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

12. Final simplification 43.6%

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

Alternative 7: 44.1% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

2. Taylor expanded in x around inf 42.8%

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

3. Final simplification 42.8%

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

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 63.4%

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

   1. sub-neg 63.4%

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

   2. +-commutative 63.4%

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

   3. flip-+ 63.4%

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

   4. sqr-neg 63.4%

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

   5. rem-square-sqrt 63.1%

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

3. Applied egg-rr 63.1%

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

4. Taylor expanded in x around inf 7.2%

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

   1. flip-- 5.8%

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

   2. clear-num 5.6%

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

   3. clear-num 5.8%

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

   4. flip-- 7.3%

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

6. Applied egg-rr 7.3%

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

7. Taylor expanded in x around 0 5.3%

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

   1. *-commutative 5.3%

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

9. Simplified 5.3%

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

10. Final simplification 5.3%

    \[\leadsto x \cdot -2 \]

Alternative 9: 11.4% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. sub-neg 63.4%

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

   2. +-commutative 63.4%

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

   3. flip-+ 63.4%

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

   4. sqr-neg 63.4%

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

   5. rem-square-sqrt 63.1%

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

3. Applied egg-rr 63.1%

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

4. Taylor expanded in x around 0 99.6%

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

   1. mul-1-neg 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 62.6%

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

   1. mul-1-neg 5.2%

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

9. Simplified 62.6%

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

10. Taylor expanded in eps around 0 11.1%

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

11. Final simplification 11.1%

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

Alternative 10: 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 63.4%

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

   1. sub-neg 63.4%

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

   2. +-commutative 63.4%

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

   3. flip-+ 63.4%

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

   4. sqr-neg 63.4%

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

   5. rem-square-sqrt 63.1%

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

3. Applied egg-rr 63.1%

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

4. Taylor expanded in x around inf 7.0%

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

5. Taylor expanded in x around 0 5.2%

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

   1. mul-1-neg 5.2%

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

7. Simplified 5.2%

   \[\leadsto \frac{\color{blue}{\left(-\varepsilon\right)} - x \cdot x}{\left(-x\right) - x} \]
8. Step-by-step derivation

   1. flip-- 0.0%

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

   2. associate-/r/ 0.0%

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

   3. sub-neg 0.0%

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

   4. distribute-neg-out 0.0%

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

   5. +-commutative 0.0%

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

   6. fma-def 0.0%

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

   7. cancel-sign-sub-inv 0.0%

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

   8. sqr-neg 0.0%

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

   9. distribute-rgt-out 0.0%

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

   10. +-commutative 0.0%

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

   11. neg-mul-1 0.0%

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

   12. metadata-eval 0.0%

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

   13. distribute-lft1-in 0.0%

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

   14. metadata-eval 0.0%

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

   15. metadata-eval 0.0%

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

   16. neg-mul-1 0.0%

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

   17. metadata-eval 0.0%

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

   18. distribute-lft1-in 0.0%

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

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x, \varepsilon\right)}{x \cdot \left(0 \cdot x\right)} \cdot \left(0 \cdot x\right)} \]
10. Step-by-step derivation

    1. mul0-lft 0.0%

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

    2. mul0-rgt 4.2%

       \[\leadsto \color{blue}{0} \]

11. Simplified 4.2%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 4.2%

    \[\leadsto 0 \]

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 2023297 
(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))))