ENA, Section 1.4, Exercise 4d

Percentage Accurate: 68.6% → 99.5%

Time: 19.5s

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	68.6%
Target	99.5%
Herbie	99.5%

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

Derivation

1. Initial program 66.6%

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

2. Applied egg-rr 66.0%

   \[\leadsto \color{blue}{\left(\sqrt{x} + {\left(x \cdot x - \varepsilon\right)}^{0.25}\right) \cdot \left(\sqrt{x} - {\left(x \cdot x - \varepsilon\right)}^{0.25}\right)} \]
   Step-by-step derivation

   [Start] 66.6%	\[ x - \sqrt{x \cdot x - \varepsilon} \]
   add-sqr-sqrt [=>] 66.5%	\[ \color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{x \cdot x - \varepsilon} \]
   add-sqr-sqrt [=>] 66.0%	\[ \sqrt{x} \cdot \sqrt{x} - \color{blue}{\sqrt{\sqrt{x \cdot x - \varepsilon}} \cdot \sqrt{\sqrt{x \cdot x - \varepsilon}}} \]
   difference-of-squares [=>] 66.0%	\[ \color{blue}{\left(\sqrt{x} + \sqrt{\sqrt{x \cdot x - \varepsilon}}\right) \cdot \left(\sqrt{x} - \sqrt{\sqrt{x \cdot x - \varepsilon}}\right)} \]
   pow1/2 [=>] 66.0%	\[ \left(\sqrt{x} + \sqrt{\color{blue}{{\left(x \cdot x - \varepsilon\right)}^{0.5}}}\right) \cdot \left(\sqrt{x} - \sqrt{\sqrt{x \cdot x - \varepsilon}}\right) \]
   sqrt-pow1 [=>] 66.2%	\[ \left(\sqrt{x} + \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(\frac{0.5}{2}\right)}}\right) \cdot \left(\sqrt{x} - \sqrt{\sqrt{x \cdot x - \varepsilon}}\right) \]
   metadata-eval [=>] 66.2%	\[ \left(\sqrt{x} + {\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.25}}\right) \cdot \left(\sqrt{x} - \sqrt{\sqrt{x \cdot x - \varepsilon}}\right) \]
   pow1/2 [=>] 66.2%	\[ \left(\sqrt{x} + {\left(x \cdot x - \varepsilon\right)}^{0.25}\right) \cdot \left(\sqrt{x} - \sqrt{\color{blue}{{\left(x \cdot x - \varepsilon\right)}^{0.5}}}\right) \]
   sqrt-pow1 [=>] 66.0%	\[ \left(\sqrt{x} + {\left(x \cdot x - \varepsilon\right)}^{0.25}\right) \cdot \left(\sqrt{x} - \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(\frac{0.5}{2}\right)}}\right) \]
   metadata-eval [=>] 66.0%	\[ \left(\sqrt{x} + {\left(x \cdot x - \varepsilon\right)}^{0.25}\right) \cdot \left(\sqrt{x} - {\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.25}}\right) \]

3. Applied egg-rr 66.3%

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

   [Start] 66.0%	\[ \left(\sqrt{x} + {\left(x \cdot x - \varepsilon\right)}^{0.25}\right) \cdot \left(\sqrt{x} - {\left(x \cdot x - \varepsilon\right)}^{0.25}\right) \]
   difference-of-squares [<=] 66.0%	\[ \color{blue}{\sqrt{x} \cdot \sqrt{x} - {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   add-sqr-sqrt [<=] 65.9%	\[ \color{blue}{x} - {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25} \]
   flip-- [=>] 65.9%	\[ \color{blue}{\frac{x \cdot x - \left({\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}\right) \cdot \left({\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}}} \]
   pow-prod-up [=>] 66.4%	\[ \frac{x \cdot x - \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(0.25 + 0.25\right)}} \cdot \left({\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   metadata-eval [=>] 66.4%	\[ \frac{x \cdot x - {\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.5}} \cdot \left({\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   pow1/2 [<=] 66.4%	\[ \frac{x \cdot x - \color{blue}{\sqrt{x \cdot x - \varepsilon}} \cdot \left({\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}\right)}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   pow-prod-up [=>] 66.1%	\[ \frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(0.25 + 0.25\right)}}}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   metadata-eval [=>] 66.1%	\[ \frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot {\left(x \cdot x - \varepsilon\right)}^{\color{blue}{0.5}}}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   pow1/2 [<=] 66.1%	\[ \frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \color{blue}{\sqrt{x \cdot x - \varepsilon}}}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   add-sqr-sqrt [<=] 66.2%	\[ \frac{x \cdot x - \color{blue}{\left(x \cdot x - \varepsilon\right)}}{x + {\left(x \cdot x - \varepsilon\right)}^{0.25} \cdot {\left(x \cdot x - \varepsilon\right)}^{0.25}} \]
   pow-prod-up [=>] 66.3%	\[ \frac{x \cdot x - \left(x \cdot x - \varepsilon\right)}{x + \color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(0.25 + 0.25\right)}}} \]

4. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{0 + \varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}} \]
   Step-by-step derivation

   [Start] 66.3%	\[ \frac{x \cdot x - \left(x \cdot x - \varepsilon\right)}{x + \sqrt{x \cdot x - \varepsilon}} \]
   associate--r- [=>] 99.5%	\[ \frac{\color{blue}{\left(x \cdot x - x \cdot x\right) + \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}} \]
   +-inverses [=>] 99.5%	\[ \frac{\color{blue}{0} + \varepsilon}{x + \sqrt{x \cdot x - \varepsilon}} \]

5. Applied egg-rr 5.5%

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

   [Start] 99.5%	\[ \frac{0 + \varepsilon}{x + \sqrt{x \cdot x - \varepsilon}} \]
   expm1-log1p-u [=>] 99.5%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0 + \varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right)} \]
   expm1-udef [=>] 5.5%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{0 + \varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1} \]
   +-lft-identity [=>] 5.5%	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1 \]

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}} \]
   Step-by-step derivation

   [Start] 5.5%	\[ e^{\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1 \]
   expm1-def [=>] 99.5%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right)} \]
   expm1-log1p [=>] 99.5%	\[ \color{blue}{\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}} \]

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6976

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

Alternative 2
Accuracy	97.5%
Cost	13764

\[\begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \]

Alternative 3
Accuracy	86.7%
Cost	7053

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-115} \lor \neg \left(x \leq 1.35 \cdot 10^{-100}\right) \land x \leq 2.9 \cdot 10^{-96}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \]

Alternative 4
Accuracy	38.6%
Cost	704

\[\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)} \]

Alternative 5
Accuracy	37.6%
Cost	320

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

Alternative 6
Accuracy	37.6%
Cost	320

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

Alternative 7
Accuracy	37.8%
Cost	320

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

Alternative 8
Accuracy	5.5%
Cost	192

\[x \cdot -2 \]

Reproduce

herbie shell --seed 2023263 
(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))))