Average Error: 16.7 → 3.7
Time: 37.5s
Precision: binary64
Cost: 77888
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\left(e^{\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} + -1\right) + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R \]
(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
    (sin phi1)
    (sin phi2)
    (*
     (cos phi2)
     (*
      (cos phi1)
      (+
       (+ (exp (log1p (* (sin lambda1) (sin lambda2)))) -1.0)
       (* (cos lambda1) (cos lambda2)))))))
  R))
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(sin(phi1), sin(phi2), (cos(phi2) * (cos(phi1) * ((exp(log1p((sin(lambda1) * sin(lambda2)))) + -1.0) + (cos(lambda1) * cos(lambda2))))))) * R;
}

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(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(phi1) * Float64(Float64(exp(log1p(Float64(sin(lambda1) * sin(lambda2)))) + -1.0) + Float64(cos(lambda1) * cos(lambda2))))))) * R)
end

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[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Exp[N[Log[1 + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

\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(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\left(e^{\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} + -1\right) + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R

Error

Derivation

  1. Initial program 16.7

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

  2. Simplified16.7

    \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
    Proof
    (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 0 points increase in error, 0 points decrease in error
    (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi2) (cos.f64 phi1)) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 6 points increase in error, 1 points decrease in error
    (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi1) (cos.f64 phi2))) (cos.f64 (-.f64 lambda1 lambda2))))) R): 0 points increase in error, 0 points decrease in error
    (*.f64 (acos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 8 points increase in error, 1 points decrease in error

  3. Applied egg-rr3.7

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

  4. Applied egg-rr3.7

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} - 1\right)} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R \]

  5. Final simplification3.7

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\left(e^{\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} + -1\right) + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R \]

Alternatives

Alternative 1
Error3.7
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \]
Alternative 2
Error3.7
Cost64960
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right) \]
Alternative 3
Error3.7
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \]
Alternative 4
Error3.7
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \]
Alternative 5
Error10.6
Cost46024
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -5.1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \end{array} \]
Alternative 6
Error10.6
Cost45768
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \end{array} \]
Alternative 7
Error10.6
Cost45640
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -4.7 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error16.9
Cost39764
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{if}\;\phi_2 \leq -0.0003:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 1.25 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{+255}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error16.9
Cost39500
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 14.6:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error10.6
Cost39496
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error24.8
Cost39236
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.00022:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \sqrt[3]{{\cos \left(\lambda_2 - \lambda_1\right)}^{3}}\right)\\ \end{array} \]
Alternative 12
Error23.2
Cost39236
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_1 \leq -1.8:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 13
Error31.3
Cost39108
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 0.00023:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \sqrt[3]{{t_0}^{3}}\right)\\ \end{array} \]
Alternative 14
Error31.3
Cost32836
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 0.0013:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \sqrt[3]{{t_0}^{3}}\right)\\ \end{array} \]
Alternative 15
Error33.4
Cost32644
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 0.00095:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \sqrt[3]{{t_0}^{3}}\right)\\ \end{array} \]
Alternative 16
Error33.3
Cost26436
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 17
Error32.6
Cost20168
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \mathbf{if}\;\phi_2 \leq -1.85 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 9.4 \cdot 10^{-148}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0 + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 18
Error41.4
Cost19784
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -7 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 7 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 19
Error40.1
Cost19652
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 20
Error36.0
Cost19648
\[R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \]
Alternative 21
Error52.1
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right) \]

Error

Reproduce

herbie shell --seed 2022331 
(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))