ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.1% → 99.4%

Time: 9.0s

Alternatives: 6

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: 62.1% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \sqrt{\varepsilon \cdot \mathsf{fma}\left(x, \frac{x}{\varepsilon}, -1\right)}} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around inf 59.8%

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

   1. unpow2 59.8%

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

   2. associate-/l* 59.9%

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

   3. fmm-def 59.9%

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

   4. metadata-eval 59.9%

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

5. Simplified 59.9%

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

   1. sub-neg 59.9%

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

   2. flip-+ 59.8%

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

   3. pow2 59.8%

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

7. Applied egg-rr 59.8%

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

   1. div-sub 59.8%

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

   2. sqr-neg 59.8%

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

   3. rem-square-sqrt 59.4%

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

   4. div-sub 59.5%

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

   5. sub-neg 59.5%

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

   6. remove-double-neg 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in x around 0 99.5%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-153) t_0 (/ eps (+ (* -0.5 (/ eps x)) (* x 2.0))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	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 <= (-5d-153)) then
        tmp = t_0
    else
        tmp = eps / (((-0.5d0) * (eps / x)) + (x * 2.0d0))
    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 <= -5e-153) {
		tmp = t_0;
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-153:
		tmp = t_0
	else:
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-153)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(-0.5 * Float64(eps / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -5e-153)
		tmp = t_0;
	else
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	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, -5e-153], t$95$0, N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.00000000000000033e-153

   1. Initial program 99.6%

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

   if -5.00000000000000033e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.0%

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

      1. flip-- 8.1%

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

      2. div-inv 8.1%

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

      3. add-sqr-sqrt 8.5%

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

      4. associate--r- 99.6%

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

      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 56.6%

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

      9. hypot-define 56.6%

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

   4. Applied egg-rr 56.6%

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

      1. *-commutative 56.6%

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

      2. +-inverses 56.6%

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

      3. +-lft-identity 56.6%

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

      4. associate-*l/ 56.8%

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

      5. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   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. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 98.4%

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

      9. mul-1-neg 98.4%

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

      10. distribute-lft-neg-in 98.4%

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

      11. distribute-rgt-neg-in 98.4%

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

      12. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in eps around 0 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-107}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.35e-107)
   (- x (sqrt (- eps)))
   (/ eps (+ (* -0.5 (/ eps x)) (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.35e-107) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.35d-107) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (((-0.5d0) * (eps / x)) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.35e-107) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.35e-107:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.35e-107)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(Float64(-0.5 * Float64(eps / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.35e-107)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / ((-0.5 * (eps / x)) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.35e-107], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.34999999999999999e-107

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0 92.7%

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

      1. neg-mul-1 92.7%

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

   5. Simplified 92.7%

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

   if 2.34999999999999999e-107 < x

   1. Initial program 24.0%

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

      1. flip-- 24.0%

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

      2. div-inv 23.9%

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

      3. add-sqr-sqrt 24.3%

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

      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 66.1%

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

      9. hypot-define 66.1%

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

   4. Applied egg-rr 66.1%

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

      1. *-commutative 66.1%

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

      2. +-inverses 66.1%

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

      3. +-lft-identity 66.1%

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

      4. associate-*l/ 66.3%

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

      5. *-lft-identity 66.3%

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

   6. Simplified 66.3%

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

   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. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 83.3%

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

      9. mul-1-neg 83.3%

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

      10. distribute-lft-neg-in 83.3%

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

      11. distribute-rgt-neg-in 83.3%

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

      12. metadata-eval 83.3%

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

   9. Simplified 83.3%

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

   10. Taylor expanded in eps around 0 83.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-107}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{-0.5 \cdot \frac{\varepsilon}{x} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps / N[(N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $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 59.7%

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

   4. associate--r- 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 80.8%

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

   9. hypot-define 80.8%

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

4. Applied egg-rr 80.8%

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

   1. *-commutative 80.8%

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

   2. +-inverses 80.8%

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

   3. +-lft-identity 80.8%

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

   4. associate-*l/ 80.8%

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

   5. *-lft-identity 80.8%

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

6. Simplified 80.8%

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

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. fma-define 0.0%

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

   4. associate-*r/ 0.0%

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

   5. *-commutative 0.0%

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

   6. *-commutative 0.0%

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

   7. unpow2 0.0%

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

   8. rem-square-sqrt 46.5%

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

   9. mul-1-neg 46.5%

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

   10. distribute-lft-neg-in 46.5%

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

   11. distribute-rgt-neg-in 46.5%

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

   12. metadata-eval 46.5%

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

9. Simplified 46.5%

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

10. Taylor expanded in eps around 0 46.5%

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

11. Final simplification 46.5%

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

Alternative 5: 44.4% 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 46.2%

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

4. Final simplification 46.2%

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

Alternative 6: 5.7% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -2.0

Julia

function code(x, eps)
	return -2.0
end

MATLAB

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

Wolfram

code[x_, eps_] := -2.0

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%

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

6. Simplified 5.5%

   \[\leadsto \color{blue}{-2} \]
7. 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 2024165 
(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))))