Average Error: 37.0 → 32.3
Time: 5.8m
Precision: 64
Internal Precision: 128
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\phi_2 \le 1.1229085965198049 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{elif}\;\phi_2 \le 4.700615090900083 \cdot 10^{-96}:\\ \;\;\;\;R \cdot \frac{\sqrt{\left(\sqrt[3]{2} \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \left({\phi_1}^{3} - {\phi_2}^{3}\right) + \sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_2 \cdot \phi_2 + \phi_1 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)}}{\sqrt{\left(\phi_1 \cdot \phi_1 + \left(\phi_1 \cdot \phi_2 + \phi_2 \cdot \phi_2\right)\right) \cdot \sqrt[3]{2}}}\\ \mathbf{elif}\;\phi_2 \le 6.178213661611158 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

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 3 regimes

  2. if phi2 < 1.1229085965198049e-150 or 4.700615090900083e-96 < phi2 < 6.178213661611158e+127

    1. Initial program 34.6

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

    3. Applied associate-*l*34.6

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

    5. Applied add-cbrt-cube34.6

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

    if 1.1229085965198049e-150 < phi2 < 4.700615090900083e-96

    1. Initial program 30.4

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

    3. Applied associate-*l*30.4

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

    5. Applied add-cbrt-cube30.5

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

    7. Applied flip3--34.6

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

    8. Applied associate-*r/36.6

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

    9. Applied cos-mult36.6

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

    10. Applied associate-*l/36.6

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

    11. Applied cbrt-div36.6

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

    12. Applied associate-*l/36.6

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

    13. Applied associate-*r/36.6

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

    14. Applied frac-add40.0

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

    15. Applied sqrt-div40.1

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

    16. Simplified31.9

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

    if 6.178213661611158e+127 < phi2

    1. Initial program 55.2

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

    3. Applied associate-*l*55.2

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

    4. Taylor expanded around 0 16.9

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

  4. Final simplification32.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \le 1.1229085965198049 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{elif}\;\phi_2 \le 4.700615090900083 \cdot 10^{-96}:\\ \;\;\;\;R \cdot \frac{\sqrt{\left(\sqrt[3]{2} \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \left({\phi_1}^{3} - {\phi_2}^{3}\right) + \sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\phi_2 \cdot \phi_2 + \phi_1 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)}}{\sqrt{\left(\phi_1 \cdot \phi_1 + \left(\phi_1 \cdot \phi_2 + \phi_2 \cdot \phi_2\right)\right) \cdot \sqrt[3]{2}}}\\ \mathbf{elif}\;\phi_2 \le 6.178213661611158 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(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: 5.8m)Debug log

sample222.0ms

Algorithm
intervals

simplify111.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

111.0ms
(* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.0b

localize60.0ms

Local error

Found 4 expressions with local error:

39.0b
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))
3.4b
(cos (/ (+ phi1 phi2) 2))
3.4b
(cos (/ (+ phi1 phi2) 2))
0.2b
(* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))

rewrite83.0ms

Algorithm
rewrite-expression-head
Counts
4 → 129
Calls

4 calls. Slowest were:

46.0ms
(* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))
23.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))
4.0ms
(cos (/ (+ phi1 phi2) 2))

series552.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

373.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))
127.0ms
(* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))
26.0ms
(cos (/ (+ phi1 phi2) 2))
25.0ms
(cos (/ (+ phi1 phi2) 2))

simplify27.8s

Counts
118 → 141
Calls

118 calls. Slowest were:

1.2s
(* (* (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))))
606.0ms
(* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))))
604.0ms
(sqrt (+ (* (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2)))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2)))) (* (* (+ lambda1 lambda2) (+ lambda1 lambda2)) (* (- (pow phi1 3) (pow phi2 3)) (- phi1 phi2)))))

prune3.0s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 24.3b

localize18.0ms

Local error

Found 4 expressions with local error:

39.0b
(sqrt (+ (* (- lambda1 lambda2) (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (* (- phi1 phi2) (- phi1 phi2))))
3.4b
(cos (/ (+ phi1 phi2) 2))
3.4b
(cos (/ (+ phi1 phi2) 2))
0.2b
(* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))

rewrite66.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

36.0ms
(sqrt (+ (* (- lambda1 lambda2) (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (* (- phi1 phi2) (- phi1 phi2))))
15.0ms
(* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))
2.0ms
(cos (/ (+ phi1 phi2) 2))

series666.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

478.0ms
(sqrt (+ (* (- lambda1 lambda2) (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (* (- phi1 phi2) (- phi1 phi2))))
150.0ms
(* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))
20.0ms
(cos (/ (+ phi1 phi2) 2))
19.0ms
(cos (/ (+ phi1 phi2) 2))

simplify20.1s

Counts
93 → 124
Calls

93 calls. Slowest were:

1.0s
(* (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2))) (* (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))))
597.0ms
(* (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2))) (* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2)))))
523.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (* (cos (/ (+ phi1 phi2) 2)) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2)))) (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (* (- (pow phi1 3) (pow phi2 3)) (- phi1 phi2)))))

prune2.8s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 24.3b

localize47.0ms

Local error

Found 4 expressions with local error:

39.0b
(sqrt (+ (* (- lambda1 lambda2) (* (cbrt (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (* (- phi1 phi2) (- phi1 phi2))))
3.4b
(cos (/ (+ phi1 phi2) 2))
3.4b
(cos (/ (+ phi1 phi2) 2))
3.4b
(cos (/ (+ phi1 phi2) 2))

rewrite81.0ms

Algorithm
rewrite-expression-head
Counts
4 → 171
Calls

4 calls. Slowest were:

64.0ms
(sqrt (+ (* (- lambda1 lambda2) (* (cbrt (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (* (- phi1 phi2) (- phi1 phi2))))
3.0ms
(cos (/ (+ phi1 phi2) 2))
3.0ms
(cos (/ (+ phi1 phi2) 2))

series490.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

436.0ms
(sqrt (+ (* (- lambda1 lambda2) (* (cbrt (* (* (cos (/ (+ phi1 phi2) 2)) (cos (/ (+ phi1 phi2) 2))) (cos (/ (+ phi1 phi2) 2)))) (* (- lambda1 lambda2) (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))

simplify44.6s

Counts
154 → 183
Calls

154 calls. Slowest were:

741.0ms
(sqrt (+ (* (* (- (pow lambda1 3) (pow lambda2 3)) (* (cbrt (* (+ (cos (+ (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2))) (cos (- (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2)))) (cos (/ (+ phi1 phi2) 2)))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))))) (* (+ phi1 phi2) (+ phi1 phi2))) (* (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (* (cbrt 2) (+ lambda1 lambda2))) (* (- (* phi1 phi1) (* phi2 phi2)) (- (* phi1 phi1) (* phi2 phi2))))))
728.0ms
(sqrt (+ (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (* (cbrt (* (+ (cos (+ (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2))) (cos (- (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2)))) (cos (/ (+ phi1 phi2) 2)))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))))) (* (+ phi1 phi2) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (* (+ lambda1 lambda2) (* (cbrt 2) (+ lambda1 lambda2))) (* (- (* phi1 phi1) (* phi2 phi2)) (- (pow phi1 3) (pow phi2 3))))))
660.0ms
(sqrt (+ (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (* (cbrt (* (+ (cos (+ (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2))) (cos (- (/ (+ phi1 phi2) 2) (/ (+ phi1 phi2) 2)))) (cos (/ (+ phi1 phi2) 2)))) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2))) (* (* (+ lambda1 lambda2) (* (cbrt 2) (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))))) (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))))))

prune5.6s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 21.9b

localize24.0ms

Local error

Found 4 expressions with local error:

39.0b
(sqrt (+ (* (- lambda1 lambda2) (* (* (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))))) (* (- phi1 phi2) (- phi1 phi2))))
3.4b
(cos (/ (+ phi1 phi2) 2))
3.4b
(cos (/ (+ phi1 phi2) 2))
3.4b
(cos (/ (+ phi1 phi2) 2))

rewrite410.0ms

Algorithm
rewrite-expression-head
Counts
4 → 675
Calls

4 calls. Slowest were:

265.0ms
(sqrt (+ (* (- lambda1 lambda2) (* (* (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))))) (* (- phi1 phi2) (- phi1 phi2))))
3.0ms
(cos (/ (+ phi1 phi2) 2))
3.0ms
(cos (/ (+ phi1 phi2) 2))

series465.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

411.0ms
(sqrt (+ (* (- lambda1 lambda2) (* (* (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))))) (* (- phi1 phi2) (- phi1 phi2))))
23.0ms
(cos (/ (+ phi1 phi2) 2))
16.0ms
(cos (/ (+ phi1 phi2) 2))
15.0ms
(cos (/ (+ phi1 phi2) 2))

simplify3.3m

Counts
658 → 687
Calls

658 calls. Slowest were:

845.0ms
(sqrt (+ (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (* (* (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2)))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (* (+ lambda1 lambda2) (* (cbrt (+ lambda1 lambda2)) (cbrt (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2)))))) (* (- (pow phi1 3) (pow phi2 3)) (- (pow phi1 3) (pow phi2 3))))))
813.0ms
(sqrt (+ (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (* (* (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2)))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (* (+ lambda1 lambda2) (* (* (cbrt (+ lambda1 lambda2)) (cbrt (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))))) (cbrt (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2)))))) (* (- (pow phi1 3) (pow phi2 3)) (- (pow phi1 3) (pow phi2 3))))))
758.0ms
(sqrt (+ (* (* (- lambda1 lambda2) (* (* (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi1 phi2) 2))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi1 phi2) 2)))))) (cbrt (* (cos (/ (+ phi1 phi2) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))))))) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))))) (* (* (cbrt (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2)))) (cbrt (+ lambda1 lambda2))) (* (- (pow phi1 3) (pow phi2 3)) (- (pow phi1 3) (pow phi2 3))))))

prune30.2s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 21.9b

regimes979.0ms

Accuracy

32.3% (10.0b remaining)

Error of 32.3b against oracle of 22.2b and baseline of 37.0b

bsearch877.0ms

end0.0ms

sample6.8s

Algorithm
intervals