Average Error: 0.9 → 0.9
Time: 17.3s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{1}{\frac{{\cos \phi_1}^{2} + \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(t_0 - \cos \phi_1\right)}{{\cos \phi_1}^{3} + {t_0}^{3}}}} \end{array} \]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\begin{array}{l}
t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{1}{\frac{{\cos \phi_1}^{2} + \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(t_0 - \cos \phi_1\right)}{{\cos \phi_1}^{3} + {t_0}^{3}}}}
\end{array}

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
↓
(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2)))))
   (+
    lambda1
    (atan2
     (*
      (cos phi2)
      (fma (sin lambda1) (cos lambda2) (- (* (cos lambda1) (sin lambda2)))))
     (/
      1.0
      (/
       (+
        (pow (cos phi1) 2.0)
        (*
         (*
          (cos phi2)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))
         (- t_0 (cos phi1))))
       (+ (pow (cos phi1) 3.0) (pow t_0 3.0))))))))
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
}
↓
double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	return lambda1 + atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), -(cos(lambda1) * sin(lambda2)))), (1.0 / ((pow(cos(phi1), 2.0) + ((cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))) * (t_0 - cos(phi1)))) / (pow(cos(phi1), 3.0) + pow(t_0, 3.0)))));
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.9

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

  2. Applied egg-rr0.9

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

  3. Applied egg-rr0.9

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

  4. Applied egg-rr0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2022129 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))