Average Error: 24.1 → 0.4
Time: 5.5s
Precision: binary64
\[\left(0 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[x - \sqrt{x \cdot x - \varepsilon} \]
↓
\[\frac{1}{\frac{x + \sqrt{x \cdot x - \varepsilon}}{\varepsilon}} \]
x - \sqrt{x \cdot x - \varepsilon}
↓
\frac{1}{\frac{x + \sqrt{x \cdot x - \varepsilon}}{\varepsilon}}

(FPCore (x eps) :precision binary64 (- x (sqrt (- (* x x) eps))))
↓
(FPCore (x eps)
 :precision binary64
 (/ 1.0 (/ (+ x (sqrt (- (* x x) eps))) eps)))
double code(double x, double eps) {
	return x - sqrt(((x * x) - eps));
}
↓
double code(double x, double eps) {
	return 1.0 / ((x + sqrt(((x * x) - eps))) / eps);
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.1
Target0.3
Herbie0.4
\[\frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}} \]

Derivation

  1. Initial program 24.1

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

  2. Applied egg-rr24.3

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

  3. Taylor expanded in x around 0 0.4

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

    \[\leadsto \frac{1}{\frac{x + \sqrt{x \cdot x - \varepsilon}}{\varepsilon}} \]

Reproduce

herbie shell --seed 2022130 
(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))))