Average Error: 17.0 → 3.8

Time: 1.9min

Precision: binary64

Cost: 58688

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

↓

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

\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R

↓

R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (*
     (* (cos phi1) (cos phi2))
     (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))))))

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1 - lambda2))) * R;
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
}

Error

The X axis uses an exponential scale — Bits error versus `R`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	10.7
Cost	52930

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -9134.866914517399:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \log \left(e^{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.780596553667369 \cdot 10^{+19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sqrt[3]{{\left(\sin \phi_1 \cdot \sin \phi_2\right)}^{3}}\right)\\ \end{array}\]

Alternative 2
Error	10.2
Cost	52930

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.027208884307120835:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 6.43302924540169 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array}\]

Alternative 3
Error	10.3
Cost	52353

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.3476736591040927 \cdot 10^{-06}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 6.43302924540169 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array}\]

Alternative 4
Error	10.3
Cost	46088

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.4959925547783745 \cdot 10^{-06} \lor \neg \left(\phi_1 \leq 6.43302924540169 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array}\]

Alternative 5
Error	17.6
Cost	39560

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq 2.6172104874922033 \cdot 10^{-214} \lor \neg \left(\phi_1 \leq 5.686091583663815 \cdot 10^{-134}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array}\]

Alternative 6
Error	17.0
Cost	39232

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

Alternative 7
Error	17.1
Cost	39432

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.179662043253059 \cdot 10^{-07} \lor \neg \left(\lambda_1 \leq 4.9186064226726405\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array}\]

Alternative 8
Error	23.3
Cost	39432

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.06083364880851677 \lor \neg \left(\phi_1 \leq 2.2623110052704576 \cdot 10^{-05}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array}\]

Alternative 9
Error	30.2
Cost	33160

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -20020833338044.004 \lor \neg \left(\phi_2 \leq 5.367813589352923 \cdot 10^{+83}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array}\]

Alternative 10
Error	36.1
Cost	32832

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

Alternative 11
Error	36.4
Cost	33032

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -29861605941705.953 \lor \neg \left(\lambda_1 \leq 0.11743439269419638\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array}\]

Alternative 12
Error	36.4
Cost	26632

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -8.036029096285144 \cdot 10^{-84} \lor \neg \left(\phi_2 \leq 2.1928564797568394 \cdot 10^{-62}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array}\]

Alternative 13
Error	41.2
Cost	26304

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

Alternative 14
Error	41.4
Cost	26504

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -29861605941705.953 \lor \neg \left(\lambda_1 \leq 0.12266396068473054\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array}\]

Alternative 15
Error	44.3
Cost	26504

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.009600956894570675 \lor \neg \left(\phi_2 \leq 0.19951728626067106\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right)\\ \end{array}\]

Alternative 16
Error	51.1
Cost	19776

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

Alternative 17
Error	52.4
Cost	13376

\[R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\]

Alternative 18
Error	61.3
Cost	64

\[0\]

Alternative 19
Error	61.7
Cost	64

\[1\]

Error

Time

Derivation

Initial program 17.0
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
Using strategy rm
Applied sub-neg_binary64_41217.0
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) \cdot R\]
Applied cos-sum_binary64_5533.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}\right) \cdot R\]
Simplified3.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R\]
Simplified3.8
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\sin \lambda_1 \cdot \left(-\sin \lambda_2\right)}\right)\right) \cdot R\]
Using strategy rm
Applied pow1_binary64_4803.8
\[\leadsto \color{blue}{{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)\right)}^{1}} \cdot R\]
Simplified3.8
\[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\]
Final simplification3.8
\[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\]

Reproduce

herbie shell --seed 2021065 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))