Average Error: 0.9 → 0.9
Time: 14.4s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t_0\\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\frac{{\cos \phi_1}^{3} + {t_1}^{3}}{\mathsf{fma}\left(\log \left(e^{t_0}\right), \cos \phi_2 \cdot \left(t_1 - \cos \phi_1\right), \cos \phi_1 \cdot \cos \phi_1\right)}} \end{array} \]
\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 \left(\lambda_1 - \lambda_2\right)\\
t_1 := \cos \phi_2 \cdot t_0\\
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\frac{{\cos \phi_1}^{3} + {t_1}^{3}}{\mathsf{fma}\left(\log \left(e^{t_0}\right), \cos \phi_2 \cdot \left(t_1 - \cos \phi_1\right), \cos \phi_1 \cdot \cos \phi_1\right)}}
\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 (- lambda1 lambda2))) (t_1 (* (cos phi2) t_0)))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (/
      (+ (pow (cos phi1) 3.0) (pow t_1 3.0))
      (fma
       (log (exp t_0))
       (* (cos phi2) (- t_1 (cos phi1)))
       (* (cos phi1) (cos phi1))))))))
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(lambda1 - lambda2);
	double t_1 = cos(phi2) * t_0;
	return lambda1 + atan2((cos(phi2) * sin(lambda1 - lambda2)), ((pow(cos(phi1), 3.0) + pow(t_1, 3.0)) / fma(log(exp(t_0)), (cos(phi2) * (t_1 - cos(phi1))), (cos(phi1) * cos(phi1)))));
}

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 flip3-+_binary640.9

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

  3. Simplified0.9

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

  4. Simplified0.9

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

  5. Applied add-log-exp_binary640.9

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

  6. Final simplification0.9

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

Reproduce

herbie shell --seed 2022005 
(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)))))))