ENA, Section 1.4, Exercise 4d

Average Error: 24.5 → 0.3

Time: 5.8s

Precision: binary64

Cost: 6976

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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 = x - sqrt(((x * x) - eps))
end function

↓

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 x - Math.sqrt(((x * x) - eps));
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	24.5
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 24.5

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

2. Applied egg-rr 24.8

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

3. Applied egg-rr 24.8

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

4. Taylor expanded in x around 0 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.0
Cost	13764

\[\begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot \left(0.125 \cdot \frac{\varepsilon}{x \cdot x} + 0.5\right)\\ \end{array} \]

Alternative 2
Error	8.5
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1.42 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{x} + 2 \cdot \frac{x}{\varepsilon}}\\ \end{array} \]

Alternative 3
Error	35.0
Cost	704

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

Alternative 4
Error	35.5
Cost	320

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

Alternative 5
Error	35.4
Cost	320

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

Alternative 6
Error	60.6
Cost	192

\[x \cdot -2 \]

Alternative 7
Error	56.6
Cost	192

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

Alternative 8
Error	60.6
Cost	128

\[-x \]

Alternative 9
Error	61.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022308 
(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))))