Average Error: 38.9 → 0.2
Time: 17.6s
Precision: binary64
Cost: 40256
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \left(0.3333333333333333 \cdot \left(1.5 \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(0.5 \cdot \phi_2\right)\right)\right)\right)\right)\right), \phi_1 - \phi_2\right) \]
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (sqrt
   (+
    (*
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
    (* (- phi1 phi2) (- phi1 phi2))))))
↓
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (*
     2.0
     (*
      0.3333333333333333
      (*
       1.5
       (fma
        (cos (* 0.5 phi2))
        (cos (* 0.5 phi1))
        (* (sin (* 0.5 phi1)) (- (sin (* 0.5 phi2)))))))))
   (- phi1 phi2))))
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * (2.0 * (0.3333333333333333 * (1.5 * fma(cos((0.5 * phi2)), cos((0.5 * phi1)), (sin((0.5 * phi1)) * -sin((0.5 * phi2)))))))), (phi1 - phi2));
}

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * sqrt(Float64(Float64(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))) * Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end
↓
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * Float64(2.0 * Float64(0.3333333333333333 * Float64(1.5 * fma(cos(Float64(0.5 * phi2)), cos(Float64(0.5 * phi1)), Float64(sin(Float64(0.5 * phi1)) * Float64(-sin(Float64(0.5 * phi2))))))))), Float64(phi1 - phi2)))
end

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(2.0 * N[(0.3333333333333333 * N[(1.5 * N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * (-N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

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 \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \left(0.3333333333333333 \cdot \left(1.5 \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(0.5 \cdot \phi_2\right)\right)\right)\right)\right)\right), \phi_1 - \phi_2\right)

Error

Derivation

  1. Initial program 38.9

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

  2. Simplified3.6

    \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
    Proof
    (*.f64 R (hypot.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (-.f64 phi1 phi2))): 0 points increase in error, 0 points decrease in error
    (*.f64 R (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))))): 164 points increase in error, 0 points decrease in error

  3. Applied egg-rr3.8

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}}\right) + \log \left(\sqrt{e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}}\right)\right)}, \phi_1 - \phi_2\right) \]

  4. Simplified3.8

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}}\right)\right)}, \phi_1 - \phi_2\right) \]
    Proof
    (*.f64 2 (log.f64 (sqrt.f64 (exp.f64 (cos.f64 (*.f64 1/2 (+.f64 phi2 phi1))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (log.f64 (sqrt.f64 (exp.f64 (cos.f64 (*.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 phi1 phi2)))))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (log.f64 (sqrt.f64 (exp.f64 (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 phi1 phi2) 1/2))))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= count-2_binary64 (+.f64 (log.f64 (sqrt.f64 (exp.f64 (cos.f64 (*.f64 (+.f64 phi1 phi2) 1/2))))) (log.f64 (sqrt.f64 (exp.f64 (cos.f64 (*.f64 (+.f64 phi1 phi2) 1/2))))))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr3.7

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(1.5 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)}\right), \phi_1 - \phi_2\right) \]

  6. Applied egg-rr0.2

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \left(0.3333333333333333 \cdot \left(1.5 \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}\right)\right)\right), \phi_1 - \phi_2\right) \]

  7. Final simplification0.2

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \left(0.3333333333333333 \cdot \left(1.5 \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(0.5 \cdot \phi_2\right)\right)\right)\right)\right)\right), \phi_1 - \phi_2\right) \]

Alternatives

Alternative 1
Error0.2
Cost33920
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(2 \cdot \left(0.3333333333333333 \cdot \left(1.5 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right)\right), \phi_1 - \phi_2\right) \]
Alternative 2
Error14.3
Cost13700
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.6 \cdot 10^{-125}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 3
Error3.6
Cost13696
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \]
Alternative 4
Error18.7
Cost13572
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]
Alternative 5
Error16.8
Cost13572
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0065:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]
Alternative 6
Error23.6
Cost13444
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.4 \cdot 10^{-37}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]
Alternative 7
Error23.9
Cost7180
\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.6 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 180000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
Alternative 8
Error24.0
Cost6916
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.48 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
Alternative 9
Error46.3
Cost968
\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 1.35 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(0.125 \cdot \left(\phi_2 \cdot \phi_2\right)\right) - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error48.2
Cost652
\[\begin{array}{l} t_0 := \phi_1 \cdot \left(-R\right)\\ \mathbf{if}\;\phi_2 \leq 3.8 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-83}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 11
Error46.3
Cost584
\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 1.2 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error49.4
Cost388
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 13
Error57.5
Cost192
\[R \cdot \lambda_1 \]
Alternative 14
Error53.6
Cost192
\[R \cdot \phi_2 \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))