ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 99.2%

Time: 9.9s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (+ x (+ x (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-154:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = eps / (x + (x + (-0.5 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(-0.5 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-154], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 98.2%

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

      1. flip-- 98.0%

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

      2. div-inv 97.9%

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

      3. add-sqr-sqrt 97.5%

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

      4. sub-neg 97.5%

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

      5. add-sqr-sqrt 97.5%

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

      6. hypot-def 97.5%

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

   3. Applied egg-rr 97.5%

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

      1. associate-*r/ 97.6%

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

      2. *-rgt-identity 97.6%

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

      3. associate--r- 99.2%

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

      4. +-inverses 99.2%

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

      5. +-lft-identity 99.2%

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

   5. Simplified 99.2%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.8%

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

      1. flip-- 8.8%

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

      2. div-inv 8.8%

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

      3. add-sqr-sqrt 8.8%

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

      4. sub-neg 8.8%

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

      5. add-sqr-sqrt 3.1%

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

      6. hypot-def 3.1%

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

   3. Applied egg-rr 3.1%

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

      1. associate-*r/ 3.1%

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

      2. *-rgt-identity 3.1%

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

      3. associate--r- 57.0%

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

      4. +-inverses 57.0%

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

      5. +-lft-identity 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 98.6%

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

      5. neg-mul-1 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in eps around 0 98.6%

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

4. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (hypot (sqrt (- eps)) x))
   (/ eps (+ x (+ x (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - Math.hypot(Math.sqrt(-eps), x);
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-154:
		tmp = x - math.hypot(math.sqrt(-eps), x)
	else:
		tmp = eps / (x + (x + (-0.5 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-154)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(-0.5 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154)
		tmp = x - hypot(sqrt(-eps), x);
	else
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-154], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 98.2%

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

      1. sub-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. add-sqr-sqrt 98.2%

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

      4. hypot-def 98.2%

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

   3. Applied egg-rr 98.2%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.8%

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

      1. flip-- 8.8%

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

      2. div-inv 8.8%

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

      3. add-sqr-sqrt 8.8%

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

      4. sub-neg 8.8%

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

      5. add-sqr-sqrt 3.1%

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

      6. hypot-def 3.1%

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

   3. Applied egg-rr 3.1%

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

      1. associate-*r/ 3.1%

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

      2. *-rgt-identity 3.1%

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

      3. associate--r- 57.0%

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

      4. +-inverses 57.0%

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

      5. +-lft-identity 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 98.6%

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

      5. neg-mul-1 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in eps around 0 98.6%

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

4. Final simplification 98.3%

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

Alternative 3: 99.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154) t_0 (/ eps (+ x (+ x (* -0.5 (/ eps x))))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (-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 <= (-2d-154)) then
        tmp = t_0
    else
        tmp = eps / (x + (x + ((-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 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-154:
		tmp = t_0
	else:
		tmp = eps / (x + (x + (-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 <= -2e-154)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(x + Float64(x + 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 <= -2e-154)
		tmp = t_0;
	else
		tmp = eps / (x + (x + (-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, -2e-154], t$95$0, N[(eps / N[(x + N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 98.2%

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

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.8%

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

      1. flip-- 8.8%

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

      2. div-inv 8.8%

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

      3. add-sqr-sqrt 8.8%

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

      4. sub-neg 8.8%

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

      5. add-sqr-sqrt 3.1%

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

      6. hypot-def 3.1%

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

   3. Applied egg-rr 3.1%

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

      1. associate-*r/ 3.1%

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

      2. *-rgt-identity 3.1%

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

      3. associate--r- 57.0%

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

      4. +-inverses 57.0%

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

      5. +-lft-identity 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 98.6%

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

      5. neg-mul-1 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in eps around 0 98.6%

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

4. Final simplification 98.3%

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

Alternative 4: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.8e-108)
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.8e-108) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (-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 <= 3.8d-108) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + ((-0.5d0) * (eps / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.8e-108) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.8e-108:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + (-0.5 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.8e-108)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(-0.5 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.8e-108)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.8e-108], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.79999999999999973e-108

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 94.5%

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

      1. neg-mul-1 94.5%

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

   4. Simplified 94.5%

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

   if 3.79999999999999973e-108 < x

   1. Initial program 23.9%

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

      1. flip-- 23.8%

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

      2. div-inv 23.8%

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

      3. add-sqr-sqrt 23.9%

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

      4. sub-neg 23.9%

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

      5. add-sqr-sqrt 19.8%

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

      6. hypot-def 19.8%

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

   3. Applied egg-rr 19.8%

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

      1. associate-*r/ 19.8%

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

      2. *-rgt-identity 19.8%

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

      3. associate--r- 64.3%

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

      4. +-inverses 64.3%

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

      5. +-lft-identity 64.3%

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

   5. Simplified 64.3%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. fma-def 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 84.4%

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

      5. neg-mul-1 84.4%

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

   8. Simplified 84.4%

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

   9. Taylor expanded in eps around 0 84.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \]

Alternative 5: 45.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. flip-- 63.1%

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

   2. div-inv 63.1%

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

   3. add-sqr-sqrt 62.9%

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

   4. sub-neg 62.9%

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

   5. add-sqr-sqrt 60.6%

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

   6. hypot-def 60.6%

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

3. Applied egg-rr 60.6%

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

   1. associate-*r/ 60.7%

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

   2. *-rgt-identity 60.7%

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

   3. associate--r- 82.7%

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

   4. +-inverses 82.7%

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

   5. +-lft-identity 82.7%

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

5. Simplified 82.7%

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

6. Taylor expanded in x around inf 0.0%

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

   1. fma-def 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 44.1%

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

   5. neg-mul-1 44.1%

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

8. Simplified 44.1%

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

9. Taylor expanded in eps around 0 44.1%

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

10. Final simplification 44.1%

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

Alternative 6: 44.4% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. flip-- 63.1%

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

   2. div-inv 63.1%

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

   3. add-sqr-sqrt 62.9%

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

   4. sub-neg 62.9%

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

   5. add-sqr-sqrt 60.6%

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

   6. hypot-def 60.6%

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

3. Applied egg-rr 60.6%

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

   1. associate-*r/ 60.7%

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

   2. *-rgt-identity 60.7%

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

   3. associate--r- 82.7%

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

   4. +-inverses 82.7%

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

   5. +-lft-identity 82.7%

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

5. Simplified 82.7%

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

6. Taylor expanded in x around inf 0.0%

   \[\leadsto \color{blue}{-0.125 \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\varepsilon}^{2}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x}} \]
7. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x} + -0.125 \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\varepsilon}^{2}}{{x}^{3}}} \]

   2. *-commutative 0.0%

      \[\leadsto \color{blue}{\frac{\varepsilon}{x} \cdot 0.5} + -0.125 \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\varepsilon}^{2}}{{x}^{3}} \]

   3. associate-*l/ 0.0%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot 0.5}{x}} + -0.125 \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\varepsilon}^{2}}{{x}^{3}} \]

   4. associate-*r/ 0.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{0.5}{x}} + -0.125 \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\varepsilon}^{2}}{{x}^{3}} \]

   5. associate-*r/ 0.0%

      \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \color{blue}{\frac{-0.125 \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\varepsilon}^{2}\right)}{{x}^{3}}} \]

   6. associate-*r* 0.0%

      \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \frac{\color{blue}{\left(-0.125 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot {\varepsilon}^{2}}}{{x}^{3}} \]

   7. associate-/l* 0.0%

      \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \color{blue}{\frac{-0.125 \cdot {\left(\sqrt{-1}\right)}^{2}}{\frac{{x}^{3}}{{\varepsilon}^{2}}}} \]

   8. unpow2 0.0%

      \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \frac{-0.125 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}{\frac{{x}^{3}}{{\varepsilon}^{2}}} \]

   9. rem-square-sqrt 38.1%

      \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \frac{-0.125 \cdot \color{blue}{-1}}{\frac{{x}^{3}}{{\varepsilon}^{2}}} \]

   10. metadata-eval 38.1%

       \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \frac{\color{blue}{0.125}}{\frac{{x}^{3}}{{\varepsilon}^{2}}} \]

   11. associate-/l* 38.1%

       \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot {\varepsilon}^{2}}{{x}^{3}}} \]

   12. associate-*r/ 38.1%

       \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \color{blue}{0.125 \cdot \frac{{\varepsilon}^{2}}{{x}^{3}}} \]

   13. unpow2 38.1%

       \[\leadsto \varepsilon \cdot \frac{0.5}{x} + 0.125 \cdot \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{{x}^{3}} \]

   14. associate-*r/ 38.6%

       \[\leadsto \varepsilon \cdot \frac{0.5}{x} + 0.125 \cdot \color{blue}{\left(\varepsilon \cdot \frac{\varepsilon}{{x}^{3}}\right)} \]

   15. *-commutative 38.6%

       \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \color{blue}{\left(\varepsilon \cdot \frac{\varepsilon}{{x}^{3}}\right) \cdot 0.125} \]

   16. associate-*l* 38.6%

       \[\leadsto \varepsilon \cdot \frac{0.5}{x} + \color{blue}{\varepsilon \cdot \left(\frac{\varepsilon}{{x}^{3}} \cdot 0.125\right)} \]

   17. distribute-lft-out 38.6%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{0.5}{x} + \frac{\varepsilon}{{x}^{3}} \cdot 0.125\right)} \]

8. Simplified 38.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{0.5}{x} + \frac{\varepsilon}{{x}^{3}} \cdot 0.125\right)} \]

9. Taylor expanded in x around inf 43.1%

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

10. Final simplification 43.1%

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

Alternative 7: 44.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

2. Taylor expanded in x around inf 43.3%

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

   1. *-commutative 43.3%

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

   2. associate-*l/ 43.3%

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

4. Simplified 43.3%

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

5. Final simplification 43.3%

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

Alternative 8: 5.4% 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 63.3%

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

   1. flip-- 63.1%

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

   2. div-inv 63.1%

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

   3. add-sqr-sqrt 62.9%

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

   4. sub-neg 62.9%

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

   5. add-sqr-sqrt 60.6%

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

   6. hypot-def 60.6%

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

3. Applied egg-rr 60.6%

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

   1. associate-*r/ 60.7%

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

   2. *-rgt-identity 60.7%

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

   3. associate--r- 82.7%

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

   4. +-inverses 82.7%

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

   5. +-lft-identity 82.7%

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

5. Simplified 82.7%

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

6. Taylor expanded in x around inf 0.0%

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

   1. fma-def 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 44.1%

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

   5. neg-mul-1 44.1%

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

8. Simplified 44.1%

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

9. Taylor expanded in eps around inf 5.5%

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

    1. *-commutative 5.5%

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

11. Simplified 5.5%

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

12. Final simplification 5.5%

    \[\leadsto x \cdot -2 \]

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 2023279 
(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))))