ENA, Section 1.4, Exercise 4d

Average Error: 24.6 → 0.3

Time: 6.9s

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	24.6
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 24.6

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

2. Applied egg-rr 0.4

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

3. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\varepsilon + 0}{x + \sqrt{x \cdot x - \varepsilon}}} \]
   Proof

   [Start] 0.4	\[ \left(\varepsilon + x \cdot \left(x - x\right)\right) \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]
   *-commutative [=>] 0.4	\[ \color{blue}{\frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \cdot \left(\varepsilon + x \cdot \left(x - x\right)\right)} \]
   associate-*l/ [=>] 0.3	\[ \color{blue}{\frac{1 \cdot \left(\varepsilon + x \cdot \left(x - x\right)\right)}{x + \sqrt{x \cdot x - \varepsilon}}} \]
   *-lft-identity [=>] 0.3	\[ \frac{\color{blue}{\varepsilon + x \cdot \left(x - x\right)}}{x + \sqrt{x \cdot x - \varepsilon}} \]
   +-inverses [=>] 0.3	\[ \frac{\varepsilon + x \cdot \color{blue}{0}}{x + \sqrt{x \cdot x - \varepsilon}} \]
   mul0-rgt [=>] 0.3	\[ \frac{\varepsilon + \color{blue}{0}}{x + \sqrt{x \cdot x - \varepsilon}} \]

4. Applied egg-rr 58.2

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1} \]

5. Simplified 0.3

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

   [Start] 58.2	\[ e^{\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1 \]
   expm1-def [=>] 0.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right)} \]
   expm1-log1p [=>] 0.3	\[ \color{blue}{\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}} \]

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.5
Cost	13764

\[\begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]

Alternative 2
Error	9.3
Cost	7053

\[\begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-114} \lor \neg \left(x \leq 2.35 \cdot 10^{-82}\right) \land x \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \end{array} \]

Alternative 3
Error	9.1
Cost	7053

\[\begin{array}{l} t_0 := \sqrt{-\varepsilon}\\ \mathbf{if}\;x \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\varepsilon}{x + t_0}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-82} \lor \neg \left(x \leq 2.15 \cdot 10^{-77}\right):\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \mathbf{else}:\\ \;\;\;\;x - t_0\\ \end{array} \]

Alternative 4
Error	35.4
Cost	320

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

Alternative 5
Error	56.7
Cost	192

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

Alternative 6
Error	61.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023057 
(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))))