Average Error: 15.7 → 15.2
Time: 17.4s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - \frac{1}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}} \cdot \frac{{0.5}^{3}}{\mathsf{hypot}\left(1, x\right)}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - \frac{1}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}} \cdot \frac{{0.5}^{3}}{\mathsf{hypot}\left(1, x\right)}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
↓
(FPCore (x)
 :precision binary64
 (/
  (*
   1.0
   (exp
    (log
     (/
      (-
       (pow (- 1.0 0.5) 3.0)
       (* (/ 1.0 (pow (hypot 1.0 x) 2.0)) (/ (pow 0.5 3.0) (hypot 1.0 x))))
      (+
       (* (- 1.0 0.5) (- 1.0 0.5))
       (* (/ 0.5 (hypot 1.0 x)) (+ (- 1.0 0.5) (/ 0.5 (hypot 1.0 x)))))))))
  (+ 1.0 (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) (1.0 * ((double) exp(((double) log((((double) (((double) pow(((double) (1.0 - 0.5)), 3.0)) - ((double) ((1.0 / ((double) pow(((double) hypot(1.0, x)), 2.0))) * (((double) pow(0.5, 3.0)) / ((double) hypot(1.0, x))))))) / ((double) (((double) (((double) (1.0 - 0.5)) * ((double) (1.0 - 0.5)))) + ((double) ((0.5 / ((double) hypot(1.0, x))) * ((double) (((double) (1.0 - 0.5)) + (0.5 / ((double) hypot(1.0, x)))))))))))))))) / ((double) (1.0 + ((double) sqrt(((double) (0.5 * ((double) (1.0 + (1.0 / ((double) hypot(1.0, x))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 15.7 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: 15.7 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: 15.2 bits

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

  6. Applied add-exp-logError: 15.2 bits

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

  8. Applied flip3--Error: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \color{blue}{\left(\frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right) \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  9. SimplifiedError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrtError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  12. Applied *-un-lft-identityError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{\color{blue}{1 \cdot 0.5}}{\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  13. Applied times-fracError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - {\color{blue}{\left(\frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{0.5}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  14. Applied unpow-prod-downError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - \color{blue}{{\left(\frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}^{3} \cdot {\left(\frac{0.5}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  15. SimplifiedError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - \color{blue}{\frac{1}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}} \cdot {\left(\frac{0.5}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  16. SimplifiedError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - \frac{1}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}} \cdot \color{blue}{\frac{{0.5}^{3}}{\mathsf{hypot}\left(1, x\right)}}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + \left(1 - 0.5\right)\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  17. Final simplificationError: 15.2 bits

    \[\leadsto \frac{1 \cdot e^{\log \left(\frac{{\left(1 - 0.5\right)}^{3} - \frac{1}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}} \cdot \frac{{0.5}^{3}}{\mathsf{hypot}\left(1, x\right)}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))