ENA, Section 1.4, Exercise 4d

Average Error: 24.5 → 1.1

Time: 10.9s

Precision: binary64

Cost: 14020

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\left(x \cdot x + \varepsilon \cdot -2\right) + \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -2e-153)
   (- x (sqrt (+ (+ (* x x) (* eps -2.0)) eps)))
   (* 0.5 (/ eps x))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - sqrt((((x * x) + (eps * -2.0)) + eps));
	} 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) :: tmp
    if ((x - sqrt(((x * x) - eps))) <= (-2d-153)) then
        tmp = x - sqrt((((x * x) + (eps * (-2.0d0))) + eps))
    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 tmp;
	if ((x - Math.sqrt(((x * x) - eps))) <= -2e-153) {
		tmp = x - Math.sqrt((((x * x) + (eps * -2.0)) + eps));
	} else {
		tmp = 0.5 * (eps / x);
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	tmp = 0
	if (x - math.sqrt(((x * x) - eps))) <= -2e-153:
		tmp = x - math.sqrt((((x * x) + (eps * -2.0)) + eps))
	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)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -2e-153)
		tmp = Float64(x - sqrt(Float64(Float64(Float64(x * x) + Float64(eps * -2.0)) + eps)));
	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)
	tmp = 0.0;
	if ((x - sqrt(((x * x) - eps))) <= -2e-153)
		tmp = x - sqrt((((x * x) + (eps * -2.0)) + eps));
	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_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e-153], N[(x - N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] + N[(eps * -2.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]]

Target

Original	24.5
Target	0.3
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -2.00000000000000008e-153

   1. Initial program 0.8

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

   2. Applied egg-rr 0.8

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

   if -2.00000000000000008e-153 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 58.5

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

   2. Taylor expanded in x around inf 1.6

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;x - \sqrt{\left(x \cdot x + \varepsilon \cdot -2\right) + \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	13764

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

Alternative 2
Error	8.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-101}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]

Alternative 3
Error	8.4
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-101}:\\ \;\;\;\;-\sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]

Alternative 4
Error	35.5
Cost	320

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

Alternative 5
Error	61.3
Cost	192

\[x - x \]

Alternative 6
Error	61.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023096 
(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))))