Average Error: 24.3 → 24.0
Time: 52.9s
Precision: binary64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
↓
\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\\ t_1 := {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\\ t_2 := \mathsf{fma}\left({\cos \lambda_2}^{2}, {\cos \lambda_1}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_2\right)\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), t_1\right)}}{\sqrt{{\left({\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \left(\frac{{\sin \lambda_1}^{3}}{\frac{t_2}{{\sin \lambda_2}^{3}}} + \frac{{\cos \lambda_2}^{3}}{t_2} \cdot {\cos \lambda_1}^{3}\right)\right), t_1\right)\right)}^{3}\right)}^{0.3333333333333333}}}\right) \end{array} \]
(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 (pow (sin (* -0.5 (- phi2 phi1))) 2.0))
        (t_2
         (-
          (fma
           (pow (cos lambda2) 2.0)
           (pow (cos lambda1) 2.0)
           (* (pow (sin lambda2) 2.0) (pow (sin lambda1) 2.0)))
          (*
           (sin lambda1)
           (* (cos lambda1) (* (cos lambda2) (sin lambda2)))))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (fma (cos phi1) (* (cos phi2) (* t_0 t_0)) t_1))
      (sqrt
       (pow
        (pow
         (-
          1.0
          (fma
           (cos phi1)
           (*
            (cos phi2)
            (+
             0.5
             (*
              -0.5
              (+
               (/ (pow (sin lambda1) 3.0) (/ t_2 (pow (sin lambda2) 3.0)))
               (* (/ (pow (cos lambda2) 3.0) t_2) (pow (cos lambda1) 3.0))))))
           t_1))
         3.0)
        0.3333333333333333)))))))
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 = pow(sin((-0.5 * (phi2 - phi1))), 2.0);
	double t_2 = fma(pow(cos(lambda2), 2.0), pow(cos(lambda1), 2.0), (pow(sin(lambda2), 2.0) * pow(sin(lambda1), 2.0))) - (sin(lambda1) * (cos(lambda1) * (cos(lambda2) * sin(lambda2))));
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (t_0 * t_0)), t_1)), sqrt(pow(pow((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * ((pow(sin(lambda1), 3.0) / (t_2 / pow(sin(lambda2), 3.0))) + ((pow(cos(lambda2), 3.0) / t_2) * pow(cos(lambda1), 3.0)))))), t_1)), 3.0), 0.3333333333333333))));
}

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 = sin(Float64(-0.5 * Float64(phi2 - phi1))) ^ 2.0
	t_2 = Float64(fma((cos(lambda2) ^ 2.0), (cos(lambda1) ^ 2.0), Float64((sin(lambda2) ^ 2.0) * (sin(lambda1) ^ 2.0))) - Float64(sin(lambda1) * Float64(cos(lambda1) * Float64(cos(lambda2) * sin(lambda2)))))
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(t_0 * t_0)), t_1)), sqrt(((Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * Float64(Float64((sin(lambda1) ^ 3.0) / Float64(t_2 / (sin(lambda2) ^ 3.0))) + Float64(Float64((cos(lambda2) ^ 3.0) / t_2) * (cos(lambda1) ^ 3.0)))))), t_1)) ^ 3.0) ^ 0.3333333333333333)))))
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[Sin[N[(-0.5 * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[Cos[lambda2], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Cos[lambda1], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Sin[lambda2], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[lambda1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Power[N[Power[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[(N[(N[Power[N[Sin[lambda1], $MachinePrecision], 3.0], $MachinePrecision] / N[(t$95$2 / N[Power[N[Sin[lambda2], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Cos[lambda2], $MachinePrecision], 3.0], $MachinePrecision] / t$95$2), $MachinePrecision] * N[Power[N[Cos[lambda1], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $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 := {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\\
t_2 := \mathsf{fma}\left({\cos \lambda_2}^{2}, {\cos \lambda_1}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_2\right)\right)\\
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), t_1\right)}}{\sqrt{{\left({\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \left(\frac{{\sin \lambda_1}^{3}}{\frac{t_2}{{\sin \lambda_2}^{3}}} + \frac{{\cos \lambda_2}^{3}}{t_2} \cdot {\cos \lambda_1}^{3}\right)\right), t_1\right)\right)}^{3}\right)}^{0.3333333333333333}}}\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 24.3

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

  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

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{{\left({\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}}}}\right) \]

  4. Applied egg-rr24.0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{{\left({\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\frac{{\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^{3} + {\left(\sin \lambda_2 \cdot \sin \lambda_1\right)}^{3}}{\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1, \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)}}\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}}}\right) \]

  5. Taylor expanded in lambda2 around inf 24.0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{{\left({\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - \left(0.5 \cdot \frac{{\sin \lambda_1}^{3} \cdot {\sin \lambda_2}^{3}}{\left({\cos \lambda_2}^{2} \cdot {\cos \lambda_1}^{2} + {\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_2 \cdot \left(\sin \lambda_2 \cdot \cos \lambda_1\right)\right)} + 0.5 \cdot \frac{{\cos \lambda_2}^{3} \cdot {\cos \lambda_1}^{3}}{\left({\cos \lambda_2}^{2} \cdot {\cos \lambda_1}^{2} + {\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_2 \cdot \left(\sin \lambda_2 \cdot \cos \lambda_1\right)\right)}\right)\right)}, {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}}}\right) \]

  6. Simplified24.0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{{\left({\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 + -0.5 \cdot \left(\frac{{\sin \lambda_1}^{3}}{\frac{\mathsf{fma}\left({\cos \lambda_2}^{2}, {\cos \lambda_1}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_1 \cdot \left(\sin \lambda_2 \cdot \cos \lambda_2\right)\right)}{{\sin \lambda_2}^{3}}} + \frac{{\cos \lambda_2}^{3}}{\mathsf{fma}\left({\cos \lambda_2}^{2}, {\cos \lambda_1}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_1 \cdot \left(\sin \lambda_2 \cdot \cos \lambda_2\right)\right)} \cdot {\cos \lambda_1}^{3}\right)\right)}, {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}}}\right) \]

  7. Final simplification24.0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{{\left({\left(1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \left(\frac{{\sin \lambda_1}^{3}}{\frac{\mathsf{fma}\left({\cos \lambda_2}^{2}, {\cos \lambda_1}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_2\right)\right)}{{\sin \lambda_2}^{3}}} + \frac{{\cos \lambda_2}^{3}}{\mathsf{fma}\left({\cos \lambda_2}^{2}, {\cos \lambda_1}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) - \sin \lambda_1 \cdot \left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_2\right)\right)} \cdot {\cos \lambda_1}^{3}\right)\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}}}\right) \]

Reproduce

herbie shell --seed 2022156 
(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))))))))))