Average Error: 0 → 0

Time: 8.1m

Precision: 64

Internal Precision: 128

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]

↓

\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]

Error

The X axis uses an exponential scale — Bits error versus `lambda1`

Derivation

Initial program 0
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
Initial simplification0
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
Final simplification0
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))

Details

start14.4s

Algorithm: intervals

setup46.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize51.0ms

Local error

Found 4 expressions with local error:

7.9b	(sin (- lambda1 lambda2))
7.6b	(cos (- lambda1 lambda2))
0.6b	(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
0.2b	(* (cos phi2) (sin (- lambda1 lambda2)))

rewrite21.0ms

Algorithm

rewrite-expression-head

Counts

4 → 57

Calls

4 calls. Slowest were:

12.0ms	(* (cos phi2) (sin (- lambda1 lambda2)))
4.0ms	(sin (- lambda1 lambda2))
4.0ms	(cos (- lambda1 lambda2))

series203.0ms

Counts

4 → 12

Calls

4 calls. Slowest were:

87.0ms	(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
56.0ms	(* (cos phi2) (sin (- lambda1 lambda2)))
31.0ms	(sin (- lambda1 lambda2))
29.0ms	(cos (- lambda1 lambda2))

simplify810.0ms

Counts

30 → 69

Calls

30 calls. Slowest were:

193.0ms	(- 2 (+ (* 1/2 (pow lambda1 2)) (* 1/2 (pow phi2 2))))
156.0ms	(- lambda1 (+ lambda2 (* 1/6 (pow lambda1 3))))
101.0ms	(- (+ 1 (* lambda2 lambda1)) (* 1/2 (pow lambda1 2)))

prune1.2s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b