ENA, Section 1.4, Exercise 4d

Average Accuracy: 61.1% → 99.6%

Time: 10.4s

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	61.1%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 61.1%

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

2. Applied egg-rr 99.4%

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

   [Start] 61.1	\[ x - \sqrt{x \cdot x - \varepsilon} \]
   flip-- [=>] 61.1	\[ \color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}} \]
   div-inv [=>] 60.9	\[ \color{blue}{\left(x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}\right) \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}}} \]
   add-sqr-sqrt [<=] 60.8	\[ \left(x \cdot x - \color{blue}{\left(x \cdot x - \varepsilon\right)}\right) \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]
   associate--r- [=>] 99.4	\[ \color{blue}{\left(\left(x \cdot x - x \cdot x\right) + \varepsilon\right)} \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]
   +-commutative [=>] 99.4	\[ \color{blue}{\left(\varepsilon + \left(x \cdot x - x \cdot x\right)\right)} \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]
   distribute-lft-out-- [=>] 99.4	\[ \left(\varepsilon + \color{blue}{x \cdot \left(x - x\right)}\right) \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]

3. Simplified 99.6%

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

   [Start] 99.4	\[ \left(\varepsilon + x \cdot \left(x - x\right)\right) \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]
   *-commutative [=>] 99.4	\[ \color{blue}{\frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \cdot \left(\varepsilon + x \cdot \left(x - x\right)\right)} \]
   associate-*l/ [=>] 99.6	\[ \color{blue}{\frac{1 \cdot \left(\varepsilon + x \cdot \left(x - x\right)\right)}{x + \sqrt{x \cdot x - \varepsilon}}} \]
   *-lft-identity [=>] 99.6	\[ \frac{\color{blue}{\varepsilon + x \cdot \left(x - x\right)}}{x + \sqrt{x \cdot x - \varepsilon}} \]
   +-inverses [=>] 99.6	\[ \frac{\varepsilon + x \cdot \color{blue}{0}}{x + \sqrt{x \cdot x - \varepsilon}} \]
   mul0-rgt [=>] 99.6	\[ \frac{\varepsilon + \color{blue}{0}}{x + \sqrt{x \cdot x - \varepsilon}} \]

4. Applied egg-rr 9.3%

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

   [Start] 99.6	\[ \frac{\varepsilon + 0}{x + \sqrt{x \cdot x - \varepsilon}} \]
   expm1-log1p-u [=>] 99.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon + 0}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right)} \]
   expm1-udef [=>] 9.3	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\varepsilon + 0}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1} \]
   +-rgt-identity [=>] 9.3	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1 \]

5. Simplified 99.6%

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

   [Start] 9.3	\[ e^{\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)} - 1 \]
   expm1-def [=>] 99.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right)} \]
   expm1-log1p [=>] 99.6	\[ \color{blue}{\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}} \]

6. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	97.7%
Cost	13764

\[\begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \]

Alternative 2
Accuracy	83.1%
Cost	7180

\[\begin{array}{l} t_0 := \sqrt{-\varepsilon}\\ t_1 := -0.5 \cdot \frac{\varepsilon}{x}\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;x - t_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-119}:\\ \;\;\;\;\frac{\varepsilon}{t_1 + x \cdot 2}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-76}:\\ \;\;\;\;\frac{\varepsilon}{x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + t_1\right)}\\ \end{array} \]

Alternative 3
Accuracy	82.5%
Cost	7052

\[\begin{array}{l} t_0 := x - \sqrt{-\varepsilon}\\ t_1 := -0.5 \cdot \frac{\varepsilon}{x}\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{\varepsilon}{t_1 + x \cdot 2}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + t_1\right)}\\ \end{array} \]

Alternative 4
Accuracy	45.9%
Cost	704

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

Alternative 5
Accuracy	45.9%
Cost	704

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

Alternative 6
Accuracy	45.2%
Cost	320

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

Alternative 7
Accuracy	5.3%
Cost	192

\[x \cdot -2 \]

Alternative 8
Accuracy	11.5%
Cost	192

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

Alternative 9
Accuracy	3.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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))))