ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.6% → 99.5%

Time: 9.4s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.6% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x (sqrt (- (pow x 2.0) eps)))))

C

double code(double x, double eps) {
	return eps / (x + sqrt((pow(x, 2.0) - eps)));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return eps / (x + Math.sqrt((Math.pow(x, 2.0) - eps)));
}

Python

def code(x, eps):
	return eps / (x + math.sqrt((math.pow(x, 2.0) - eps)))

Julia

function code(x, eps)
	return Float64(eps / Float64(x + sqrt(Float64((x ^ 2.0) - eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (x + sqrt(((x ^ 2.0) - eps)));
end

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{\mathsf{fma}\left(x, x, -\varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(\frac{\varepsilon \cdot \frac{1}{x}}{x} \cdot \frac{\varepsilon}{x}\right) + \frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-154)
   (- x (sqrt (fma x x (- eps))))
   (+ (* 0.125 (* (/ (* eps (/ 1.0 x)) x) (/ eps x))) (* (/ eps x) 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-154) {
		tmp = x - sqrt(fma(x, x, -eps));
	} else {
		tmp = (0.125 * (((eps * (1.0 / 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-154)
		tmp = Float64(x - sqrt(fma(x, x, Float64(-eps))));
	else
		tmp = Float64(Float64(0.125 * Float64(Float64(Float64(eps * Float64(1.0 / 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-154], N[(x - N[Sqrt[N[(x * x + (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(N[(N[(eps * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(\frac{\varepsilon \cdot \frac{1}{x}}{x} \cdot \frac{\varepsilon}{x}\right) + \frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-154)
     t_0
     (+ (* 0.125 (* (/ (* eps (/ 1.0 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-154) {
		tmp = t_0;
	} else {
		tmp = (0.125 * (((eps * (1.0 / x)) / x) * (eps / 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) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-2d-154)) then
        tmp = t_0
    else
        tmp = (0.125d0 * (((eps * (1.0d0 / x)) / x) * (eps / x))) + ((eps / x) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-154) {
		tmp = t_0;
	} else {
		tmp = (0.125 * (((eps * (1.0 / x)) / x) * (eps / x))) + ((eps / x) * 0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-154:
		tmp = t_0
	else:
		tmp = (0.125 * (((eps * (1.0 / 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-154)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.125 * Float64(Float64(Float64(eps * Float64(1.0 / x)) / x) * Float64(eps / x))) + Float64(Float64(eps / x) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -2e-154)
		tmp = t_0;
	else
		tmp = (0.125 * (((eps * (1.0 / x)) / x) * (eps / x))) + ((eps / x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-154], t$95$0, N[(N[(0.125 * N[(N[(N[(eps * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-110}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{x} + 2 \cdot \frac{x}{\varepsilon}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.85e-110)
   (- x (sqrt (- eps)))
   (/ 1.0 (+ (/ -0.5 x) (* 2.0 (/ x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.85e-110) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.85d-110) then
        tmp = x - sqrt(-eps)
    else
        tmp = 1.0d0 / (((-0.5d0) / x) + (2.0d0 * (x / eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.85e-110) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.85e-110:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = 1.0 / ((-0.5 / x) + (2.0 * (x / eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.85e-110)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(1.0 / Float64(Float64(-0.5 / x) + Float64(2.0 * Float64(x / eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.85e-110)
		tmp = x - sqrt(-eps);
	else
		tmp = 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.85e-110], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-0.5 / x), $MachinePrecision] + N[(2.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 44.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-0.5}{x} + 2 \cdot \frac{x}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ 1.0 (+ (/ -0.5 x) (* 2.0 (/ x eps)))))

C

double code(double x, double eps) {
	return 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 / (((-0.5d0) / x) + (2.0d0 * (x / eps)))
end function

Java

public static double code(double x, double eps) {
	return 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
}

Python

def code(x, eps):
	return 1.0 / ((-0.5 / x) + (2.0 * (x / eps)))

Julia

function code(x, eps)
	return Float64(1.0 / Float64(Float64(-0.5 / x) + Float64(2.0 * Float64(x / eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 / ((-0.5 / x) + (2.0 * (x / eps)));
end

Wolfram

code[x_, eps_] := N[(1.0 / N[(N[(-0.5 / x), $MachinePrecision] + N[(2.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 43.7% 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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 11.3% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 3.5% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x

Julia

function code(x, eps)
	return x
end

MATLAB

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

Wolfram

code[x_, eps_] := x

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

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 2024006 
(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))))