Average Error: 37.1 → 29.0
Time: 2.8m
Precision: 64
Internal Precision: 128
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 5.453093445784098 \cdot 10^{+306}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Error

Bits error versus R

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. Split input into 2 regimes

  2. if (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))) < 5.453093445784098e+306

    1. Initial program 1.9

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

    if 5.453093445784098e+306 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 60.6

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt60.6

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

    4. Taylor expanded around 0 47.1

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification29.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 5.453093445784098 \cdot 10^{+306}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))

Details

Time bar (total: 2.7m)Debug log

sample220.0ms

Algorithm
intervals

simplify120.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

120.0ms
(* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 39.7b

localize41.0ms

Local error

Found 4 expressions with local error:

38.6b
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))
5.5b
(cos (/ (+ phi1 phi2) 2))
5.5b
(cos (/ (+ phi1 phi2) 2))
0.2b
(* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 111
Calls

4 calls. Slowest were:

20.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))
5.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))
2.0ms
(cos (/ (+ phi1 phi2) 2))

series554.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

411.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))
104.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))
23.0ms
(cos (/ (+ phi1 phi2) 2))
17.0ms
(cos (/ (+ phi1 phi2) 2))

simplify21.5s

Counts
92 → 123
Calls

92 calls. Slowest were:

790.0ms
(sqrt (+ (* (* (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2)))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (+ lambda1 lambda2)) (* (- (pow phi1 3) (pow phi2 3)) (- (pow phi1 3) (pow phi2 3))))))
630.0ms
(sqrt (+ (* (* (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (+ phi1 phi2) (+ phi1 phi2))) (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (* (- (* phi1 phi1) (* phi2 phi2)) (- (* phi1 phi1) (* phi2 phi2))))))
575.0ms
(sqrt (+ (* (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2)))) (* (+ phi1 phi2) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (* (+ lambda1 lambda2) (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2)))) (* (- (* phi1 phi1) (* phi2 phi2)) (- (pow phi1 3) (pow phi2 3))))))

prune3.1s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 28.6b

localize24.0ms

Local error

Found 4 expressions with local error:

38.6b
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cbrt (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
5.5b
(cos (/ (+ phi1 phi2) 2))
5.5b
(cos (/ (+ phi1 phi2) 2))
5.5b
(cos (/ (+ phi1 phi2) 2))

rewrite123.0ms

Algorithm
rewrite-expression-head
Counts
4 → 171
Calls

4 calls. Slowest were:

97.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cbrt (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
4.0ms
(cos (/ (+ phi1 phi2) 2))
4.0ms
(cos (/ (+ phi1 phi2) 2))

series524.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

470.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cbrt (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
24.0ms
(cos (/ (+ phi1 phi2) 2))
15.0ms
(cos (/ (+ phi1 phi2) 2))
14.0ms
(cos (/ (+ phi1 phi2) 2))

simplify46.0s

Counts
154 → 183
Calls

154 calls. Slowest were:

815.0ms
(sqrt (+ (* (* (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cbrt (* (+ (cos (+ (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2))) (cos (- (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2)))) (cos (/ (+ phi1 phi2) 2)))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2))) (* (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (cbrt 2)) (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))))))
711.0ms
(sqrt (+ (* (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cbrt (* (+ (cos (+ (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2))) (cos (- (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2)))) (cos (/ (+ phi1 phi2) 2)))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (* (+ lambda1 lambda2) (cbrt 2)) (* (- (pow phi1 3) (pow phi2 3)) (- (pow phi1 3) (pow phi2 3))))))
642.0ms
(sqrt (+ (* (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cbrt (* (+ (cos (+ (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2))) (cos (- (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2)))) (cos (/ (+ phi1 phi2) 2)))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2))) (* (cbrt 2) (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))))))

prune5.7s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 28.3b

localize32.0ms

Local error

Found 4 expressions with local error:

38.6b
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
5.5b
(cos (/ (+ phi1 phi2) 2))
5.5b
(cos (/ (+ phi1 phi2) 2))
5.5b
(cos (/ (+ phi1 phi2) 2))

rewrite211.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

195.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
3.0ms
(cos (/ (+ phi1 phi2) 2))
3.0ms
(cos (/ (+ phi1 phi2) 2))

series418.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

364.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
20.0ms
(cos (/ (+ phi1 phi2) 2))
19.0ms
(cos (/ (+ phi1 phi2) 2))
15.0ms
(cos (/ (+ phi1 phi2) 2))

simplify24.2s

Counts
82 → 111
Calls

82 calls. Slowest were:

768.0ms
(sqrt (+ (* (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (+ lambda1 lambda2) (* (- (pow phi1 3) (pow phi2 3)) (- (pow phi1 3) (pow phi2 3))))))
562.0ms
(sqrt (+ (* (* (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2)))) (* (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2)))) (* (- phi1 phi2) (- (pow phi1 3) (pow phi2 3))))))
516.0ms
(sqrt (+ (* (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2))) (* (* (+ lambda1 lambda2) (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2)))) (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))))))

prune2.9s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 28.3b

localize51.0ms

Local error

Found 4 expressions with local error:

38.6b
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))))
38.6b
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
5.5b
(cos (/ (+ phi1 phi2) 2))
5.5b
(cos (/ (+ phi1 phi2) 2))

rewrite232.0ms

Algorithm
rewrite-expression-head
Counts
4 → 170
Calls

4 calls. Slowest were:

196.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
22.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))))
3.0ms
(cos (/ (+ phi1 phi2) 2))

series901.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

533.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))))
323.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (- phi1 phi2) (- phi1 phi2))))
23.0ms
(cos (/ (+ phi1 phi2) 2))
22.0ms
(cos (/ (+ phi1 phi2) 2))

simplify42.3s

Counts
152 → 182
Calls

152 calls. Slowest were:

684.0ms
(sqrt (+ (* (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2))) (* (+ lambda1 lambda2) (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))))))
642.0ms
(sqrt (+ (* (* (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (* (* (cbrt (cos (/ (+ phi1 phi2) 2))) (cbrt (cos (/ (+ phi1 phi2) 2)))) (cbrt (cos (/ (+ phi1 phi2) 2)))))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2)))) (* (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2)))) (* (- phi1 phi2) (- (pow phi1 3) (pow phi2 3))))))
611.0ms
(sqrt (+ (* (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))) (+ lambda1 lambda2)) (* (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2)) (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))))))

prune6.8s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 28.3b

regimes1.7s

Accuracy

63.9% (4.6b remaining)

Error of 29.0b against oracle of 24.4b and baseline of 37.1b

bsearch5.0ms

end0.0ms

sample6.8s

Algorithm
intervals