Average Error: 24.5 → 0.3
Time: 3.5s
Precision: binary64
\[\left(0 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[x - \sqrt{x \cdot x - \varepsilon} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right) \]
(FPCore (x eps) :precision binary64 (- x (sqrt (- (* x x) eps))))
↓
(FPCore (x eps)
 :precision binary64
 (expm1 (log1p (/ eps (+ x (sqrt (- (* x x) eps)))))))
double code(double x, double eps) {
	return x - sqrt(((x * x) - eps));
}
↓
double code(double x, double eps) {
	return expm1(log1p((eps / (x + sqrt(((x * x) - eps))))));
}

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}
↓
public static double code(double x, double eps) {
	return Math.expm1(Math.log1p((eps / (x + Math.sqrt(((x * x) - eps))))));
}

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))
↓
def code(x, eps):
	return math.expm1(math.log1p((eps / (x + math.sqrt(((x * x) - eps))))))

function code(x, eps)
	return Float64(x - sqrt(Float64(Float64(x * x) - eps)))
end
↓
function code(x, eps)
	return expm1(log1p(Float64(eps / Float64(x + sqrt(Float64(Float64(x * x) - eps))))))
end

code[x_, eps_] := N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(Exp[N[Log[1 + N[(eps / N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

x - \sqrt{x \cdot x - \varepsilon}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right)

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.5
Target0.3
Herbie0.3
\[\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}} \]

Derivation

  1. Initial program 24.5

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

  2. Applied egg-rr0.3

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

  3. Applied egg-rr0.4

    \[\leadsto \frac{\left(x \cdot x - x \cdot x\right) + \varepsilon}{\color{blue}{\mathsf{fma}\left({\left(x \cdot x - \varepsilon\right)}^{0.25}, {\left(x \cdot x - \varepsilon\right)}^{0.25}, x\right)}} \]

  4. Applied egg-rr0.3

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right)} \]

  5. Final simplification0.3

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}\right)\right) \]

Reproduce

herbie shell --seed 2022150 
(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))))