ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.5% → 98.5%

Time: 5.6s

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: 61.5% 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: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-152)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (fma 0.125 (* (/ eps (* x x)) (/ eps x)) (* (/ eps x) 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-152) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = fma(0.125, ((eps / (x * x)) * (eps / x)), ((eps / x) * 0.5));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-152)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = fma(0.125, Float64(Float64(eps / Float64(x * x)) * Float64(eps / x)), Float64(Float64(eps / x) * 0.5));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-152], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.6%

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

      1. flip-- 97.4%

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

      2. div-inv 97.1%

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

      3. add-sqr-sqrt 96.8%

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

      4. sub-neg 96.8%

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

      5. add-sqr-sqrt 96.8%

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

      6. hypot-def 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. associate-*r/ 96.9%

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

      2. *-rgt-identity 96.9%

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

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

   1. Initial program 5.8%

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

   2. Taylor expanded in x around inf 89.1%

      \[\leadsto \color{blue}{0.125 \cdot \frac{{\varepsilon}^{2}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. fma-def 89.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]

      2. unpow2 89.1%

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

   4. Simplified 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   5. Step-by-step derivation

      1. unpow3 89.1%

         \[\leadsto \mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{\color{blue}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

      2. times-frac 100.0%

         \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-152)
   (- x (hypot (sqrt (- eps)) x))
   (fma 0.125 (* (/ eps (* x x)) (/ eps x)) (* (/ eps x) 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-152) {
		tmp = x - hypot(sqrt(-eps), x);
	} else {
		tmp = fma(0.125, ((eps / (x * x)) * (eps / x)), ((eps / x) * 0.5));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-152)
		tmp = Float64(x - hypot(sqrt(Float64(-eps)), x));
	else
		tmp = fma(0.125, Float64(Float64(eps / Float64(x * x)) * Float64(eps / x)), Float64(Float64(eps / x) * 0.5));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-152], N[(x - N[Sqrt[N[Sqrt[(-eps)], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.6%

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

      1. sub-neg 97.6%

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

      2. +-commutative 97.6%

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

      3. add-sqr-sqrt 97.6%

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

      4. hypot-def 97.7%

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

   3. Applied egg-rr 97.7%

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

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

   1. Initial program 5.8%

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

   2. Taylor expanded in x around inf 89.1%

      \[\leadsto \color{blue}{0.125 \cdot \frac{{\varepsilon}^{2}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. fma-def 89.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]

      2. unpow2 89.1%

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

   4. Simplified 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   5. Step-by-step derivation

      1. unpow3 89.1%

         \[\leadsto \mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{\color{blue}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

      2. times-frac 100.0%

         \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-152)
     t_0
     (fma 0.125 (* (/ eps (* x x)) (/ eps x)) (* (/ eps x) 0.5)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-152) {
		tmp = t_0;
	} else {
		tmp = fma(0.125, ((eps / (x * x)) * (eps / x)), ((eps / x) * 0.5));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -2e-152)
		tmp = t_0;
	else
		tmp = fma(0.125, Float64(Float64(eps / Float64(x * x)) * Float64(eps / x)), Float64(Float64(eps / x) * 0.5));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-152], t$95$0, N[(0.125 * N[(N[(eps / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.6%

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

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

   1. Initial program 5.8%

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

   2. Taylor expanded in x around inf 89.1%

      \[\leadsto \color{blue}{0.125 \cdot \frac{{\varepsilon}^{2}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. fma-def 89.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]

      2. unpow2 89.1%

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

   4. Simplified 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   5. Step-by-step derivation

      1. unpow3 89.1%

         \[\leadsto \mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{\color{blue}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

      2. times-frac 100.0%

         \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 98.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-152) t_0 (/ eps (+ (/ -0.5 (/ x eps)) (* x 2.0))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-152) {
		tmp = t_0;
	} else {
		tmp = eps / ((-0.5 / (x / eps)) + (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 <= (-2d-152)) then
        tmp = t_0
    else
        tmp = eps / (((-0.5d0) / (x / eps)) + (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 <= -2e-152) {
		tmp = t_0;
	} else {
		tmp = eps / ((-0.5 / (x / eps)) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-152:
		tmp = t_0
	else:
		tmp = eps / ((-0.5 / (x / eps)) + (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 <= -2e-152)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(-0.5 / Float64(x / eps)) + 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 <= -2e-152)
		tmp = t_0;
	else
		tmp = eps / ((-0.5 / (x / eps)) + (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, -2e-152], t$95$0, N[(eps / N[(N[(-0.5 / N[(x / eps), $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))) < -2.00000000000000013e-152

   1. Initial program 97.6%

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

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

   1. Initial program 5.8%

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

      1. flip-- 5.8%

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

      2. div-inv 5.8%

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

      3. add-sqr-sqrt 5.9%

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

      4. sub-neg 5.9%

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

      5. add-sqr-sqrt 2.7%

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

      6. hypot-def 2.7%

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

   3. Applied egg-rr 2.7%

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

      1. associate-*r/ 2.7%

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

      2. *-rgt-identity 2.7%

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

      3. associate--r- 48.5%

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

      4. +-inverses 48.5%

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

      5. +-lft-identity 48.5%

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

   5. Simplified 48.5%

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

   6. Taylor expanded in x around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. metadata-eval 100.0%

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

      9. associate-*r/ 100.0%

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

      10. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. clear-num 100.0%

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

      5. un-div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 98.6%

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

Alternative 5: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-120}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.9e-120)
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (* (/ eps x) -0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.9e-120) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.9d-120) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + ((eps / x) * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.9e-120) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.9e-120:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + ((eps / x) * -0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.9e-120)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(Float64(eps / x) * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.9e-120)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + ((eps / x) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.9e-120], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9e-120

   1. Initial program 97.1%

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

   2. Taylor expanded in x around 0 94.0%

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

      1. neg-mul-1 94.0%

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

   4. Simplified 94.0%

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

   if 2.9e-120 < x

   1. Initial program 28.6%

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

      1. flip-- 28.5%

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

      2. div-inv 28.5%

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

      3. add-sqr-sqrt 28.5%

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

      4. sub-neg 28.5%

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

      5. add-sqr-sqrt 26.1%

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

      6. hypot-def 26.1%

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

   3. Applied egg-rr 26.1%

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

      1. associate-*r/ 26.1%

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

      2. *-rgt-identity 26.1%

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

      3. associate--r- 62.3%

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

      4. +-inverses 62.3%

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

      5. +-lft-identity 62.3%

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

   5. Simplified 62.3%

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

   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. +-commutative 0.0%

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

      2. associate-*r/ 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 79.0%

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

      5. *-commutative 79.0%

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

      6. associate-*r* 79.0%

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

      7. metadata-eval 79.0%

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

      8. associate-*r/ 79.0%

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

      9. *-commutative 79.0%

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

   8. Simplified 79.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-120}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \]

Alternative 6: 45.7% 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 60.7%

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

   1. flip-- 60.6%

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

   2. div-inv 60.4%

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

   3. add-sqr-sqrt 60.2%

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

   4. sub-neg 60.2%

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

   5. add-sqr-sqrt 58.9%

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

   6. hypot-def 58.9%

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

3. Applied egg-rr 58.9%

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

   1. associate-*r/ 59.0%

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

   2. *-rgt-identity 59.0%

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

   3. associate--r- 78.8%

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

   4. +-inverses 78.8%

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

   5. +-lft-identity 78.8%

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

5. Simplified 78.8%

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

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. +-commutative 0.0%

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

   2. associate-*r/ 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 46.4%

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

   5. *-commutative 46.4%

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

   6. associate-*r* 46.4%

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

   7. metadata-eval 46.4%

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

   8. associate-*r/ 46.4%

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

   9. *-commutative 46.4%

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

8. Simplified 46.4%

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

9. Final simplification 46.4%

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

Alternative 7: 44.9% 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 60.7%

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

2. Taylor expanded in x around inf 45.6%

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

3. Final simplification 45.6%

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

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 60.7%

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

   1. flip-- 60.6%

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

   2. div-inv 60.4%

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

   3. add-sqr-sqrt 60.2%

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

   4. sub-neg 60.2%

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

   5. add-sqr-sqrt 58.9%

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

   6. hypot-def 58.9%

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

3. Applied egg-rr 58.9%

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

   1. associate-*r/ 59.0%

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

   2. *-rgt-identity 59.0%

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

   3. associate--r- 78.8%

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

   4. +-inverses 78.8%

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

   5. +-lft-identity 78.8%

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

5. Simplified 78.8%

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

6. Taylor expanded in x around inf 0.0%

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

   1. *-commutative 0.0%

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

   2. fma-def 0.0%

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

   3. associate-*r/ 0.0%

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

   4. unpow2 0.0%

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

   5. rem-square-sqrt 46.4%

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

   6. *-commutative 46.4%

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

   7. associate-*r* 46.4%

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

   8. metadata-eval 46.4%

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

   9. associate-*r/ 46.4%

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

   10. *-commutative 46.4%

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

8. Simplified 46.4%

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

9. Taylor expanded in eps around inf 5.5%

   \[\leadsto \color{blue}{-2 \cdot x} \]
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 2023257 
(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))))