Average Error: 24.3 → 23.4
Time: 35.0s
Precision: binary64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
↓
\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\\ t_1 := \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(t_1 \cdot t_1, t_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right), {\left(\sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)\right)\right)}}\right) \end{array} \]
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (+
      (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
      (*
       (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
       (sin (/ (- lambda1 lambda2) 2.0)))))
    (sqrt
     (-
      1.0
      (+
       (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
       (*
        (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
        (sin (/ (- lambda1 lambda2) 2.0))))))))))
↓
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda2 lambda1) 2.0)))
        (t_1 (cbrt (* (sin lambda2) (sin lambda1)))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (fma
        (cos phi1)
        (* (cos phi2) (* t_0 t_0))
        (pow (sin (* -0.5 (- phi2 phi1))) 2.0)))
      (sqrt
       (expm1
        (log1p
         (-
          1.0
          (fma
           (cos phi1)
           (*
            (cos phi2)
            (-
             0.5
             (* 0.5 (fma (* t_1 t_1) t_1 (* (cos lambda2) (cos lambda1))))))
           (pow
            (+
             (* (sin (* phi2 -0.5)) (cos (* phi1 0.5)))
             (* (cos (* phi2 -0.5)) (sin (* phi1 0.5))))
            2.0)))))))))))
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))), sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))))));
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda2 - lambda1) / 2.0));
	double t_1 = cbrt((sin(lambda2) * sin(lambda1)));
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (t_0 * t_0)), pow(sin((-0.5 * (phi2 - phi1))), 2.0))), sqrt(expm1(log1p((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * fma((t_1 * t_1), t_1, (cos(lambda2) * cos(lambda1)))))), pow(((sin((phi2 * -0.5)) * cos((phi1 * 0.5))) + (cos((phi2 * -0.5)) * sin((phi1 * 0.5)))), 2.0))))))));
}

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))))), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))) * sin(Float64(Float64(lambda1 - lambda2) / 2.0)))))))))
end
↓
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda2 - lambda1) / 2.0))
	t_1 = cbrt(Float64(sin(lambda2) * sin(lambda1)))
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(t_0 * t_0)), (sin(Float64(-0.5 * Float64(phi2 - phi1))) ^ 2.0))), sqrt(expm1(log1p(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * fma(Float64(t_1 * t_1), t_1, Float64(cos(lambda2) * cos(lambda1)))))), (Float64(Float64(sin(Float64(phi2 * -0.5)) * cos(Float64(phi1 * 0.5))) + Float64(cos(Float64(phi2 * -0.5)) * sin(Float64(phi1 * 0.5)))) ^ 2.0)))))))))
end

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(Exp[N[Log[1 + N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$1 + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)
↓
\begin{array}{l}
t_0 := \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\\
t_1 := \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\\
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(t_1 \cdot t_1, t_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right), {\left(\sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)\right)\right)}}\right)
\end{array}

Error

Derivation

  1. Initial program 24.3

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

  2. Simplified24.3

    \[\leadsto \color{blue}{R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}\right)} \]

  3. Applied egg-rr24.3

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)\right)\right)}}}\right) \]

  4. Applied egg-rr24.0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \cos \lambda_2 \cdot \cos \lambda_1\right)}\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)\right)\right)}}\right) \]

  5. Applied egg-rr24.1

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \cos \lambda_2 \cdot \cos \lambda_1\right)\right), {\color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)\right)} - 1\right)}}^{2}\right)\right)\right)}}\right) \]

  6. Applied egg-rr23.4

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \cos \lambda_2 \cdot \cos \lambda_1\right)\right), {\color{blue}{\left(\sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(\left(-\phi_1\right) \cdot -0.5\right) + \cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(\left(-\phi_1\right) \cdot -0.5\right)\right)}}^{2}\right)\right)\right)}}\right) \]

  7. Final simplification23.4

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}, \cos \lambda_2 \cdot \cos \lambda_1\right)\right), {\left(\sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)\right)\right)}}\right) \]

Reproduce

herbie shell --seed 2022210 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))