Average Error: 0.0 → 0.0
Time: 1.3s
Precision: binary64
\[\sqrt{1 - x \cdot x}\]
↓
\[\sqrt{\log \left(e^{1 - x \cdot x}\right)}\]
\sqrt{1 - x \cdot x}
↓
\sqrt{\log \left(e^{1 - x \cdot x}\right)}
double code(double x) {
	return ((double) sqrt(((double) (1.0 - ((double) (x * x))))));
}
↓
double code(double x) {
	return ((double) sqrt(((double) log(((double) exp(((double) (1.0 - ((double) (x * x))))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 0.0 bits

    \[\sqrt{1 - x \cdot x}\]
  2. Using strategy rm

  3. Applied add-log-expError: 0.0 bits

    \[\leadsto \sqrt{1 - \color{blue}{\log \left(e^{x \cdot x}\right)}}\]

  4. Applied add-log-expError: 0.0 bits

    \[\leadsto \sqrt{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{x \cdot x}\right)}\]

  5. Applied diff-logError: 0.0 bits

    \[\leadsto \sqrt{\color{blue}{\log \left(\frac{e^{1}}{e^{x \cdot x}}\right)}}\]

  6. SimplifiedError: 0.0 bits

    \[\leadsto \sqrt{\log \color{blue}{\left(e^{1 - x \cdot x}\right)}}\]

  7. Final simplificationError: 0.0 bits

    \[\leadsto \sqrt{\log \left(e^{1 - x \cdot x}\right)}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))