Average Error: 24.0 → 13.5
Time: 56.1s
Precision: binary64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
↓
\[\begin{array}{l} t_0 := \sin \left(\frac{\phi_2}{2}\right)\\ t_1 := \sin \left(\frac{\phi_1}{2}\right)\\ t_2 := \cos \left(\frac{\phi_1}{2}\right)\\ t_3 := t_0 \cdot t_2\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := t_4 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ t_6 := \cos \left(\frac{\phi_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t_4, t_5, {\left(\mathsf{fma}\left(t_1, t_6, -t_3\right) + \mathsf{fma}\left(-t_0, t_2, t_3\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t_4, t_5, {\left(t_1 \cdot t_6 - t_3\right)}^{2}\right)}}\right) \end{array} \]
(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 (/ phi2 2.0)))
        (t_1 (sin (/ phi1 2.0)))
        (t_2 (cos (/ phi1 2.0)))
        (t_3 (* t_0 t_2))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (* t_4 (* (cos phi1) (cos phi2))))
        (t_6 (cos (/ phi2 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (fma t_4 t_5 (pow (+ (fma t_1 t_6 (- t_3)) (fma (- t_0) t_2 t_3)) 2.0)))
      (sqrt (- 1.0 (fma t_4 t_5 (pow (- (* t_1 t_6) t_3) 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((phi2 / 2.0));
	double t_1 = sin((phi1 / 2.0));
	double t_2 = cos((phi1 / 2.0));
	double t_3 = t_0 * t_2;
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = t_4 * (cos(phi1) * cos(phi2));
	double t_6 = cos((phi2 / 2.0));
	return R * (2.0 * atan2(sqrt(fma(t_4, t_5, pow((fma(t_1, t_6, -t_3) + fma(-t_0, t_2, t_3)), 2.0))), sqrt((1.0 - fma(t_4, t_5, pow(((t_1 * t_6) - t_3), 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(phi2 / 2.0))
	t_1 = sin(Float64(phi1 / 2.0))
	t_2 = cos(Float64(phi1 / 2.0))
	t_3 = Float64(t_0 * t_2)
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = Float64(t_4 * Float64(cos(phi1) * cos(phi2)))
	t_6 = cos(Float64(phi2 / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(fma(t_4, t_5, (Float64(fma(t_1, t_6, Float64(-t_3)) + fma(Float64(-t_0), t_2, t_3)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_4, t_5, (Float64(Float64(t_1 * t_6) - t_3) ^ 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[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$4 * t$95$5 + N[Power[N[(N[(t$95$1 * t$95$6 + (-t$95$3)), $MachinePrecision] + N[((-t$95$0) * t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$4 * t$95$5 + N[Power[N[(N[(t$95$1 * t$95$6), $MachinePrecision] - t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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{\phi_2}{2}\right)\\
t_1 := \sin \left(\frac{\phi_1}{2}\right)\\
t_2 := \cos \left(\frac{\phi_1}{2}\right)\\
t_3 := t_0 \cdot t_2\\
t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\
t_5 := t_4 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\
t_6 := \cos \left(\frac{\phi_2}{2}\right)\\
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t_4, t_5, {\left(\mathsf{fma}\left(t_1, t_6, -t_3\right) + \mathsf{fma}\left(-t_0, t_2, t_3\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t_4, t_5, {\left(t_1 \cdot t_6 - t_3\right)}^{2}\right)}}\right)
\end{array}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 24.0

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

  2. Simplified24.0

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

  3. Applied egg-rr23.5

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

  4. Applied egg-rr13.5

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

  5. Final simplification13.5

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

Reproduce

herbie shell --seed 2022133 
(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))))))))))