Average Error: 0 → 0
Time: 8.0m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1} + \lambda_1\]

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0

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

  2. Final simplification0

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

Reproduce

herbie shell --seed 2019016 
(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

Time bar (total: 8.0m)Debug log

sample15.8s

Algorithm
intervals
Results
2204×(pre true 80)
1948×(body exit 10240)
196×(body real 1280)
46×(body real 640)
11×(body real 320)
3×(body real 160)

simplify15.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
15.0ms
(+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize76.0ms

Local error

Found 4 expressions with local error:

7.9b
(cos (- lambda1 lambda2))
7.7b
(sin (- lambda1 lambda2))
0.6b
(+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))
0.1b
(* (cos phi2) (sin (- lambda1 lambda2)))

rewrite43.0ms

Algorithm
rewrite-expression-head
Rules
6×add-log-exp
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×sub-neg
3×associate-*l*
3×associate-*r*
3×distribute-lft-in
3×cos-diff
3×distribute-rgt-in
2×associate-+r+
2×cos-sum
1×flip-+
1×cbrt-unprod
1×*-commutative
1×sin-diff
1×prod-exp
1×pow-prod-down
1×flip3-+
1×sum-log
1×+-commutative
Counts
4 → 54
Calls
4 calls:
Slowest
25.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
9.0ms
(+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))
4.0ms
(cos (- lambda1 lambda2))
3.0ms
(sin (- lambda1 lambda2))

series552.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
451.0ms
(+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))
65.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
20.0ms
(sin (- lambda1 lambda2))
16.0ms
(cos (- lambda1 lambda2))

simplify599.0ms

Counts
33 → 66
Calls
33 calls:
Slowest
112.0ms
(- 2 (+ (* 1/2 (pow phi2 2)) (* 1/2 (pow phi1 2))))
100.0ms
(- lambda1 (+ lambda2 (* 1/6 (pow lambda1 3))))
63.0ms
(* (* (* (cos phi2) (cos phi2)) (cos phi2)) (* (* (sin (- lambda1 lambda2)) (sin (- lambda1 lambda2))) (sin (- lambda1 lambda2))))
61.0ms
(- (+ 1 (* lambda2 lambda1)) (* 1/2 (pow lambda1 2)))
43.0ms
(* (cos phi2) (* (cos lambda1) (sin (- lambda2))))

prune957.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample7.7m

Algorithm
intervals
Results
68232×(pre true 80)
60231×(body exit 10240)
5402×(body real 1280)
2067×(body real 640)
416×(body real 320)
96×(body real 160)
20×(body real 80)