Hyperbolic arc-cosine

Average Accuracy: 51.4% → 99.4%

Time: 4.7s

Precision: binary64

Cost: 6848

Math

\[\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right)\right)\right) \]

↓

\[\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (log.f64 (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) 1)))))

↓

(FPCore (x) :precision binary64 (log.f64 (+.f64 x (-.f64 x (/.f64 1/2 x)))))

Derivation

1. Initial program 51.4%

   \[\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right)\right)\right) \]

2. Taylor expanded in x around inf 99.4%

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, x\right)\right)\right)}\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)}\right)\right) \]
   Proof

   [Start] 99.4	\[ \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]
   associate-*r/ [=>] 99.4	\[ \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(x, \color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, 1\right), x\right)}\right)\right)\right) \]
   metadata-eval [=>] 99.4	\[ \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\frac{1}{2}}, x\right)\right)\right)\right) \]

4. Final simplification 99.4%

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	6592

\[\mathsf{log.f64}\left(\mathsf{+.f64}\left(x, x\right)\right) \]

Alternative 2
Accuracy	3.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023144 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))