Average Error: 24.2 → 0.3
Time: 3.1s
Precision: binary64
\[\left(0 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[x - \sqrt{x \cdot x - \varepsilon} \]
↓
\[\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}} \]
(FPCore (x eps) :precision binary64 (- x (sqrt (- (* x x) eps))))
↓
(FPCore (x eps) :precision binary64 (/ 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 eps / (x + sqrt(((x * x) - eps)));
}

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

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)));
}

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))
↓
def code(x, eps):
	return 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 Float64(eps / Float64(x + sqrt(Float64(Float64(x * x) - eps))))
end

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

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]

x - \sqrt{x \cdot x - \varepsilon}
↓
\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.2
Target0.3
Herbie0.3
\[\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}} \]

Derivation

  1. Initial program 24.2

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

  2. Applied flip--_binary6424.3

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

  3. Simplified0.3

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

  4. Simplified13.8

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

  5. Applied hypot-udef_binary6413.8

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2022129 
(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))))