Average Error: 37.0 → 0.1
Time: 44.4s
Precision: 64
Internal Precision: 128
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]

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. Initial program 37.0

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

  2. Simplified3.7

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

  3. Taylor expanded around -inf 3.7

    \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]
  4. Using strategy rm

  5. Applied distribute-rgt-in3.7

    \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2} + \phi_2 \cdot \frac{1}{2}\right)}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

  6. Applied cos-sum0.1

    \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

  7. Taylor expanded around -inf 0.1

    \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

  8. Final simplification0.1

    \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(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: 42.9s)Debug log

sample274.0ms

Algorithm
intervals

simplify175.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

175.0ms
(* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))))

prune30.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 3.8b

localize52.0ms

Local error

Found 4 expressions with local error:

3.4b
(cos (/ (+ phi2 phi1) 2))
0.2b
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
0.1b
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
0.0b
(hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

10.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
5.0ms
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
4.0ms
(cos (/ (+ phi2 phi1) 2))

series944.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

479.0ms
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
354.0ms
(hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2))
93.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
17.0ms
(cos (/ (+ phi2 phi1) 2))

simplify1.6s

Counts
24 → 65
Calls

24 calls. Slowest were:

295.0ms
(+ (/ (* lambda2 (* phi1 phi2)) (* (pow lambda1 2) (cos (* 1/2 (+ phi1 phi2))))) (+ (* lambda2 (cos (* 1/2 (+ phi1 phi2)))) (/ (* phi1 phi2) (* lambda1 (cos (* 1/2 (+ phi1 phi2)))))))
241.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2)))
234.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))))

prune964.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 3.7b

localize35.0ms

Local error

Found 4 expressions with local error:

3.4b
(cos (* 1/2 (+ phi1 phi2)))
0.2b
(* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2))))
0.1b
(* (hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2)) R)
0.0b
(hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

17.0ms
(* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2))))
6.0ms
(cos (* 1/2 (+ phi1 phi2)))
5.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2)) R)

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

558.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2)) R)
322.0ms
(hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2))
111.0ms
(* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2))))
25.0ms
(cos (* 1/2 (+ phi1 phi2)))

simplify1.7s

Counts
26 → 67
Calls

26 calls. Slowest were:

267.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (* 1/2 (+ phi1 phi2))) (cos (* 1/2 (+ phi1 phi2)))) (cos (* 1/2 (+ phi1 phi2)))))
246.0ms
(+ (/ (* lambda2 (* phi1 phi2)) (* (pow lambda1 2) (cos (* 1/2 (+ phi1 phi2))))) (+ (* lambda2 (cos (* 1/2 (+ phi1 phi2)))) (/ (* phi1 phi2) (* lambda1 (cos (* 1/2 (+ phi1 phi2)))))))
216.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (* 1/2 (+ phi1 phi2))))

prune995.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.1b

localize34.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (sin (* phi1 1/2)) (sin (* phi2 1/2)))
0.2b
(* (- lambda1 lambda2) (- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2)))))
0.1b
(- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2))))
0.1b
(* (cos (* phi1 1/2)) (cos (* phi2 1/2)))

rewrite132.0ms

Algorithm
rewrite-expression-head
Counts
4 → 97
Calls

4 calls. Slowest were:

86.0ms
(* (- lambda1 lambda2) (- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2)))))
29.0ms
(- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2))))
9.0ms
(* (cos (* phi1 1/2)) (cos (* phi2 1/2)))

series468.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

245.0ms
(* (- lambda1 lambda2) (- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2)))))
100.0ms
(- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2))))
61.0ms
(* (cos (* phi1 1/2)) (cos (* phi2 1/2)))
61.0ms
(* (sin (* phi1 1/2)) (sin (* phi2 1/2)))

simplify10.4s

Counts
72 → 109
Calls

72 calls. Slowest were:

693.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (- (+ (cos (+ (* phi1 1/2) (* phi2 1/2))) (cos (- (* phi1 1/2) (* phi2 1/2)))) (- (cos (- (* phi1 1/2) (* phi2 1/2))) (cos (+ (* phi1 1/2) (* phi2 1/2))))))
527.0ms
(* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (+ (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2)))))
519.0ms
(* (- lambda1 lambda2) (fma (- (sin (* phi2 1/2))) (sin (* phi1 1/2)) (* (sin (* phi2 1/2)) (sin (* phi1 1/2)))))

prune2.1s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (sin (* 1/2 phi2)) (sin (* 1/2 phi1)))
0.2b
(* (- lambda1 lambda2) (- (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1)))))
0.1b
(- (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1))))
0.1b
(* (cos (* 1/2 phi2)) (cos (* 1/2 phi1)))

rewrite113.0ms

Algorithm
rewrite-expression-head
Counts
4 → 97
Calls

4 calls. Slowest were:

65.0ms
(* (- lambda1 lambda2) (- (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1)))))
25.0ms
(- (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1))))
14.0ms
(* (cos (* 1/2 phi2)) (cos (* 1/2 phi1)))

series509.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

272.0ms
(* (- lambda1 lambda2) (- (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1)))))
101.0ms
(- (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1))))
82.0ms
(* (sin (* 1/2 phi2)) (sin (* 1/2 phi1)))
53.0ms
(* (cos (* 1/2 phi2)) (cos (* 1/2 phi1)))

simplify11.3s

Counts
72 → 109
Calls

72 calls. Slowest were:

716.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (- (+ (cos (+ (* 1/2 phi2) (* 1/2 phi1))) (cos (- (* 1/2 phi2) (* 1/2 phi1)))) (- (cos (- (* 1/2 phi2) (* 1/2 phi1))) (cos (+ (* 1/2 phi2) (* 1/2 phi1))))))
662.0ms
(* (+ (* lambda1 lambda1) (+ (* lambda2 lambda2) (* lambda1 lambda2))) (+ (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1)))))
652.0ms
(* (- (pow lambda1 3) (pow lambda2 3)) (- (* (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1))) (* (cos (* 1/2 phi2)) (cos (* 1/2 phi1)))) (* (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1))) (* (sin (* 1/2 phi2)) (sin (* 1/2 phi1))))))

prune2.1s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

regimes611.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.1b and baseline of 0.1b

bsearch4.0ms

end0.0ms

sample7.2s

Algorithm
intervals