Average Error: 15.4 → 14.9
Time: 4.5s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)}
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (/
  (- 0.5 (/ 0.5 (hypot 1.0 x)))
  (+ 1.0 (log (exp (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x))))))))))
double code(double x) {
	return ((double) (1.0 - ((double) sqrt(((double) (0.5 * ((double) (1.0 + (1.0 / ((double) hypot(1.0, x)))))))))));
}
↓
double code(double x) {
	return (((double) (0.5 - (0.5 / ((double) hypot(1.0, x))))) / ((double) (1.0 + ((double) log(((double) exp(((double) sqrt(((double) (0.5 * ((double) (1.0 + (1.0 / ((double) hypot(1.0, 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: 15.4 bits

    \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
  2. Using strategy rm

  3. Applied flip--Error: 15.4 bits

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

  4. SimplifiedError: 14.9 bits

    \[\leadsto \frac{\color{blue}{1 \cdot 1 - 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  5. Taylor expanded around 0 Error: 14.9 bits

    \[\leadsto \frac{\color{blue}{0.5 - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  6. SimplifiedError: 14.9 bits

    \[\leadsto \frac{\color{blue}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  7. Using strategy rm

  8. Applied add-log-expError: 14.9 bits

    \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \color{blue}{\log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)}}\]

  9. Final simplificationError: 14.9 bits

    \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))