Spherical law of cosines

Average Error: 16.8 → 3.8

Time: 36.1s

Precision: binary64

Cost: 71232

Math

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

↓

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

FPCore

(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
 (*
  (acos
   (fma
    (cos phi2)
    (*
     (cos phi1)
     (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1))))
    (* (sin phi1) (sin phi2))))
  R))

C

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 acos(fma(cos(phi2), (cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1)))), (sin(phi1) * sin(phi2)))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

↓

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(fma(cos(phi2), Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1)))), Float64(sin(phi1) * sin(phi2)))) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

↓

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 16.8

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

2. Applied egg-rr 3.8

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

3. Taylor expanded in phi1 around 0 3.8

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

4. Simplified 3.8

   \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
   Proof
   (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (fma.f64 (sin.f64 lambda2) (sin.f64 lambda1) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
   (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 3 points increase in error, 5 points decrease in error
   (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
   (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2))))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
   (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
   (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
   (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (Rewrite=> *-commutative_binary64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
   (acos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))))) (*.f64 (sin.f64 phi1) (sin.f64 phi2))))): 2 points increase in error, 5 points decrease in error

5. Final simplification 3.8

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

Alternatives

Alternative 1
Error	3.8
Cost	64960

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

Alternative 2
Error	3.8
Cost	58688

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

Alternative 3
Error	10.4
Cost	58436

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4.4 \cdot 10^{-24}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.02 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]

Alternative 4
Error	10.3
Cost	45768

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.65 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.3
Cost	45640

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 9.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	17.3
Cost	39632

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 7.6 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{+259}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{+299}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	10.3
Cost	39496

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	25.4
Cost	39236

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.26 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	23.8
Cost	39236

\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ \mathbf{if}\;\phi_2 \leq 7.6 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 10
Error	31.5
Cost	39108

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -30:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 11
Error	32.5
Cost	39108

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 1.14 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 12
Error	33.1
Cost	32964

\[\begin{array}{l} t_0 := \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{+56}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 13
Error	40.1
Cost	32836

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.00048:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_2 \cdot t_0\right)\\ \end{array} \]

Alternative 14
Error	36.1
Cost	32832

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

Alternative 15
Error	38.6
Cost	26436

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0 + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 16
Error	49.1
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.00046:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 17
Error	46.2
Cost	26304

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

Alternative 18
Error	51.4
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \phi_1\right)\\ \end{array} \]

Alternative 19
Error	50.5
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 20
Error	47.5
Cost	19904

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

Alternative 21
Error	52.3
Cost	13376

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

Reproduce

herbie shell --seed 2022326 
(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))