ENA, Section 1.4, Exercise 4d

Average Error: 24.7 → 1.5

Time: 7.9s

Precision: binary64

Cost: 13764

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -2e-151) t_0 (* 0.5 (/ eps x)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-151) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

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

↓

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-151)) then
        tmp = t_0
    else
        tmp = 0.5d0 * (eps / x)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -2e-151) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -2e-151:
		tmp = t_0
	else:
		tmp = 0.5 * (eps / x)
	return tmp

Julia

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

↓

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -2e-151)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(eps / x));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -2e-151)
		tmp = t_0;
	else
		tmp = 0.5 * (eps / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

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-151], t$95$0, N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	24.7
Target	0.3
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-151

   1. Initial program 0.9

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

   if -1.9999999999999999e-151 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 57.9

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

   2. Taylor expanded in x around inf 2.3

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

4. Final simplification 1.5

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

Alternatives

Alternative 1
Error	9.4
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-81}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]

Alternative 2
Error	35.3
Cost	320

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

Alternative 3
Error	61.2
Cost	192

\[x - x \]

Alternative 4
Error	61.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))