Average Error: 0.5 → 0.2
Time: 1.1min
Precision: binary64
Cost: 20800
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\sqrt[3]{0.25} + \frac{\sqrt[3]{0.25}}{x} \cdot \left(\frac{0.0763888888888889}{x} + 0.16666666666666666\right)\right) \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\sqrt[3]{0.25} + \frac{\sqrt[3]{0.25}}{x} \cdot \left(\frac{0.0763888888888889}{x} + 0.16666666666666666\right)\right) \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}
(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (-
  x
  (*
   (+
    (cbrt 0.25)
    (* (/ (cbrt 0.25) x) (+ (/ 0.0763888888888889 x) 0.16666666666666666)))
   (cbrt (+ (/ 0.125 x) (+ (/ 0.0625 (* x x)) 0.5))))))
double code(double x) {
	return sqrt(x - 1.0) * sqrt(x);
}
↓
double code(double x) {
	return x - ((cbrt(0.25) + ((cbrt(0.25) / x) * ((0.0763888888888889 / x) + 0.16666666666666666))) * cbrt((0.125 / x) + ((0.0625 / (x * x)) + 0.5)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.2
Cost832
\[x - \left(\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)\right)\]
Alternative 2
Error0.3
Cost448
\[x - \left(\frac{0.125}{x} + 0.5\right)\]
Alternative 3
Error0.5
Cost192
\[x + -0.5\]
Alternative 4
Error1.2
Cost64
\[x\]

Error

Time

Derivation

  1. Initial program 0.5

    \[\sqrt{x - 1} \cdot \sqrt{x}\]

  2. Taylor expanded around inf 0.2

    \[\leadsto \color{blue}{x - \left(0.125 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.5\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{x - \left(\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt_binary640.2

    \[\leadsto x - \color{blue}{\left(\sqrt[3]{\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)} \cdot \sqrt[3]{\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)}\right) \cdot \sqrt[3]{\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)}}\]

  6. Simplified0.2

    \[\leadsto x - \color{blue}{\left(\sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)} \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\right)} \cdot \sqrt[3]{\frac{0.125}{x} + \left(0.5 + \frac{0.0625}{x \cdot x}\right)}\]

  7. Simplified0.2

    \[\leadsto x - \left(\sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)} \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\right) \cdot \color{blue}{\sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}}\]

  8. Taylor expanded around inf 0.2

    \[\leadsto x - \color{blue}{\left(0.16666666666666666 \cdot \left({0.25}^{0.3333333333333333} \cdot \frac{1}{x}\right) + \left(0.0763888888888889 \cdot \left({0.25}^{0.3333333333333333} \cdot \frac{1}{{x}^{2}}\right) + {0.25}^{0.3333333333333333}\right)\right)} \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\]

  9. Simplified0.2

    \[\leadsto x - \color{blue}{\left(\sqrt[3]{0.25} + \frac{\sqrt[3]{0.25}}{x} \cdot \left(0.16666666666666666 + \frac{0.0763888888888889}{x}\right)\right)} \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\]

  10. Taylor expanded around 0 0.2

    \[\leadsto x - \left(\sqrt[3]{0.25} + \color{blue}{\left(0.0763888888888889 \cdot \frac{\sqrt[3]{0.25}}{{x}^{2}} + 0.16666666666666666 \cdot \frac{\sqrt[3]{0.25}}{x}\right)}\right) \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\]

  11. Simplified0.2

    \[\leadsto x - \left(\sqrt[3]{0.25} + \color{blue}{\frac{\sqrt[3]{0.25}}{x} \cdot \left(\frac{0.0763888888888889}{x} + 0.16666666666666666\right)}\right) \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\]

  12. Simplified0.2

    \[\leadsto \color{blue}{x - \left(\sqrt[3]{0.25} + \frac{\sqrt[3]{0.25}}{x} \cdot \left(\frac{0.0763888888888889}{x} + 0.16666666666666666\right)\right) \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}}\]

  13. Final simplification0.2

    \[\leadsto x - \left(\sqrt[3]{0.25} + \frac{\sqrt[3]{0.25}}{x} \cdot \left(\frac{0.0763888888888889}{x} + 0.16666666666666666\right)\right) \cdot \sqrt[3]{\frac{0.125}{x} + \left(\frac{0.0625}{x \cdot x} + 0.5\right)}\]

Reproduce

herbie shell --seed 2021065 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))