\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))
double code(double x) {
	return 1.0 - sqrt(0.5 * (1.0 + (1.0 / hypot(1.0, x))));
}

Reproduce

Please include this information when filing a bug report:

herbie shell --seed 2021060 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

Backtrace

Failed to produce proof for (*.f64 (neg.f64 (*.f64 (cbrt.f64 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))))) (cbrt.f64 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))))))) (cbrt.f64 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))) to (neg.f64 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))LC
for-loop/data/pavpan/nightlies/herbie/egg-generate-proof/src/core/simplify.rkt1678
(unnamed)/data/pavpan/nightlies/herbie/egg-generate-proof/src/core/simplify.rkt1496
egraph-add-exprs/data/pavpan/nightlies/herbie/egg-generate-proof/egg-herbie/main.rkt1540
with-egraph/data/pavpan/nightlies/herbie/egg-generate-proof/egg-herbie/main.rkt740
(unnamed)/data/pavpan/nightlies/herbie/egg-generate-proof/src/core/simplify.rkt790
simplify!/data/pavpan/nightlies/herbie/egg-generate-proof/src/mainloop.rkt2360
run-iter!/data/pavpan/nightlies/herbie/egg-generate-proof/src/mainloop.rkt3600
run-improve/data/pavpan/nightlies/herbie/egg-generate-proof/src/mainloop.rkt3770
(unnamed)/data/pavpan/nightlies/herbie/egg-generate-proof/src/sandbox.rkt546
run/opt/racket-7.9/share/pkgs/profile-lib/main.rkt392
profile-thunk/opt/racket-7.9/share/pkgs/profile-lib/main.rkt90
(unnamed)/opt/racket-7.9/collects/racket/engine.rkt4224