Average Error: 0 → 0

Time: 7.9m

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
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
Simplified0
\[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1}\]
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 2019010 +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

sample18.3s

Algorithm: intervals

simplify8.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

8.0ms

(+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))

prune60.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize67.0ms

Local error

Found 4 expressions with local error:

7.1b	(sin (- lambda1 lambda2))
6.9b	(cos (- lambda1 lambda2))
0.6b	(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
0.2b	(* (cos phi2) (sin (- lambda1 lambda2)))

rewrite20.0ms

Algorithm

rewrite-expression-head

Rules

6×	`add-cube-cbrt`
6×	`add-exp-log`
6×	`sin-sum`
6×	`add-cbrt-cube`
6×	`*-un-lft-identity`
6×	`pow1`
6×	`add-sqr-sqrt`
4×	`add-log-exp`
4×	`log1p-expm1-u`
4×	`sub-neg`
4×	`expm1-log1p-u`
3×	`associate-l`
3×	`associate-r`
2×	`distribute-lft-in`
2×	`cos-sum`
2×	`distribute-rgt-in`
1×	`fma-udef`
1×	`cbrt-unprod`
1×	`*-commutative`
1×	`sin-diff`
1×	`prod-exp`
1×	`pow-prod-down`
1×	`cos-diff`

Counts

4 → 57

Calls

4 calls:

Slowest

11.0ms	(* (cos phi2) (sin (- lambda1 lambda2)))
4.0ms	(sin (- lambda1 lambda2))
3.0ms	(cos (- lambda1 lambda2))
0.0ms	(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))

series193.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

89.0ms	(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
55.0ms	(* (cos phi2) (sin (- lambda1 lambda2)))
30.0ms	(sin (- lambda1 lambda2))
20.0ms	(cos (- lambda1 lambda2))

simplify661.0ms

Counts

30 → 69

Calls

30 calls:

Slowest

119.0ms	(- 2 (+ (* 1/2 (pow lambda1 2)) (* 1/2 (pow phi2 2))))
114.0ms	(- lambda1 (+ lambda2 (* 1/6 (pow lambda1 3))))
96.0ms	(* (* (* (cos phi2) (cos phi2)) (cos phi2)) (* (* (sin (- lambda1 lambda2)) (sin (- lambda1 lambda2))) (sin (- lambda1 lambda2))))
58.0ms	(* (* (cos lambda1) (sin (- lambda2))) (cos phi2))
49.0ms	(* (* (cos lambda1) (sin (- lambda2))) (cos phi2))

prune1.0s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample7.5m

Algorithm: intervals