Average Error: 37.5 → 33.2
Time: 2.5m
Precision: 64
Internal Precision: 128
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -5.701044550682903 \cdot 10^{+154}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\\ \end{array}\]

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) < -5.701044550682903e+154

    1. Initial program 60.8

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

    2. Initial simplification60.8

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

    3. Taylor expanded around 0 38.9

      \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R\]

    if -5.701044550682903e+154 < (- lambda1 lambda2)

    1. Initial program 31.9

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

    2. Initial simplification31.9

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

    4. Applied associate-*l*31.9

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

  4. Final simplification33.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -5.701044550682903 \cdot 10^{+154}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(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: 2.3m)Debug log

start308.0ms

Algorithm
intervals

setup521.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.0b

localize64.0ms

Local error

Found 4 expressions with local error:

36.6b
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
3.0b
(cos (/ (+ phi2 phi1) 2))
3.0b
(cos (/ (+ phi2 phi1) 2))
0.2b
(* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))

rewrite85.0ms

Algorithm
rewrite-expression-head
Counts
4 → 129
Calls

4 calls. Slowest were:

46.0ms
(* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))
24.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
3.0ms
(cos (/ (+ phi2 phi1) 2))

series831.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

683.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
97.0ms
(* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))
26.0ms
(cos (/ (+ phi2 phi1) 2))
25.0ms
(cos (/ (+ phi2 phi1) 2))

simplify28.6s

Counts
118 → 141
Calls

118 calls. Slowest were:

1.3s
(* (* (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))) (* (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))
554.0ms
(sqrt (+ (* (* (- phi1 phi2) (- (* phi1 phi1) (* phi2 phi2))) (+ lambda1 lambda2)) (* (+ phi1 phi2) (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))))
534.0ms
(* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi2 phi1) 2))))

prune3.5s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 30.2b

localize27.0ms

Local error

Found 4 expressions with local error:

36.6b
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (- lambda1 lambda2) (* (cos (/ (+ phi2 phi1) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))))
3.0b
(cos (/ (+ phi2 phi1) 2))
3.0b
(cos (/ (+ phi2 phi1) 2))
0.2b
(* (cos (/ (+ phi2 phi1) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

17.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (- lambda1 lambda2) (* (cos (/ (+ phi2 phi1) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))))
16.0ms
(* (cos (/ (+ phi2 phi1) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))
2.0ms
(cos (/ (+ phi2 phi1) 2))

series871.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

683.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (- lambda1 lambda2) (* (cos (/ (+ phi2 phi1) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))))
140.0ms
(* (cos (/ (+ phi2 phi1) 2)) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))
24.0ms
(cos (/ (+ phi2 phi1) 2))
24.0ms
(cos (/ (+ phi2 phi1) 2))

simplify22.0s

Counts
93 → 124
Calls

93 calls. Slowest were:

1.3s
(* (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))) (* (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))
881.0ms
(sqrt (+ (* (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))) (* (+ lambda1 lambda2) (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))))) (* (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2)) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (* (cos (/ (+ phi2 phi1) 2)) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi2 phi1) 2))))))))
562.0ms
(* (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))) (* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2)))))

prune2.9s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 30.2b

localize21.0ms

Local error

Found 4 expressions with local error:

36.6b
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
36.6b
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
3.0b
(cos (/ (+ phi2 phi1) 2))
3.0b
(cos (/ (+ phi2 phi1) 2))

rewrite68.0ms

Algorithm
rewrite-expression-head
Counts
4 → 170
Calls

4 calls. Slowest were:

26.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
24.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
3.0ms
(cos (/ (+ phi2 phi1) 2))

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

624.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
525.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))
24.0ms
(cos (/ (+ phi2 phi1) 2))
17.0ms
(cos (/ (+ phi2 phi1) 2))

simplify40.5s

Counts
152 → 182
Calls

152 calls. Slowest were:

627.0ms
(sqrt (+ (* (* (- (* phi1 phi1) (* phi2 phi2)) (- (pow phi1 3) (pow phi2 3))) (+ lambda1 lambda2)) (* (* (+ phi1 phi2) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2)))) (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))))
601.0ms
(sqrt (+ (* (* (- phi1 phi2) (- (pow phi1 3) (pow phi2 3))) (+ lambda1 lambda2)) (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))))
528.0ms
(sqrt (+ (* (* (- (pow phi1 3) (pow phi2 3)) (- (* phi1 phi1) (* phi2 phi2))) (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (+ lambda1 lambda2))) (* (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ phi1 phi2)) (* (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi2 phi1) 2))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2)))))))

prune6.6s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 30.2b

localize19.0ms

Local error

Found 4 expressions with local error:

36.6b
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (exp (log (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))))
29.4b
(log (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))
3.0b
(cos (/ (+ phi2 phi1) 2))
3.0b
(cos (/ (+ phi2 phi1) 2))

rewrite134.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

103.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (exp (log (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))))
22.0ms
(log (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))
2.0ms
(cos (/ (+ phi2 phi1) 2))

series689.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

528.0ms
(sqrt (+ (* (- phi1 phi2) (- phi1 phi2)) (exp (log (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))))))))
131.0ms
(log (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))))
15.0ms
(cos (/ (+ phi2 phi1) 2))
14.0ms
(cos (/ (+ phi2 phi1) 2))

simplify22.0s

Counts
102 → 137
Calls

102 calls. Slowest were:

691.0ms
(sqrt (+ (* (* (- (pow phi1 3) (pow phi2 3)) (- (pow phi1 3) (pow phi2 3))) (exp (log (* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))))))) (* (* (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2))) (+ (* phi1 phi1) (+ (* phi2 phi2) (* phi1 phi2)))) (exp (log (* (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi2 phi1) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi2 phi1) 2)))))))))
621.0ms
(log (* (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2))) (* (- (pow lambda1 3) (pow lambda2 3)) (cos (/ (+ phi2 phi1) 2)))))
608.0ms
(log (* (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2)))))

prune3.2s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 30.2b

regimes879.0ms

Accuracy

52.2% (3.8b remaining)

Error of 33.2b against oracle of 29.4b and baseline of 37.5b

bsearch9.0ms