Average Error: 0.2 → 0.2
Time: 44.1s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1}\right)}\]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1}\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r64927 = lambda1;
        double r64928 = theta;
        double r64929 = sin(r64928);
        double r64930 = delta;
        double r64931 = sin(r64930);
        double r64932 = r64929 * r64931;
        double r64933 = phi1;
        double r64934 = cos(r64933);
        double r64935 = r64932 * r64934;
        double r64936 = cos(r64930);
        double r64937 = sin(r64933);
        double r64938 = r64937 * r64936;
        double r64939 = r64934 * r64931;
        double r64940 = cos(r64928);
        double r64941 = r64939 * r64940;
        double r64942 = r64938 + r64941;
        double r64943 = asin(r64942);
        double r64944 = sin(r64943);
        double r64945 = r64937 * r64944;
        double r64946 = r64936 - r64945;
        double r64947 = atan2(r64935, r64946);
        double r64948 = r64927 + r64947;
        return r64948;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r64949 = lambda1;
        double r64950 = theta;
        double r64951 = sin(r64950);
        double r64952 = delta;
        double r64953 = sin(r64952);
        double r64954 = r64951 * r64953;
        double r64955 = phi1;
        double r64956 = cos(r64955);
        double r64957 = r64954 * r64956;
        double r64958 = cos(r64952);
        double r64959 = sin(r64955);
        double r64960 = r64956 * r64953;
        double r64961 = cos(r64950);
        double r64962 = r64960 * r64961;
        double r64963 = fma(r64959, r64958, r64962);
        double r64964 = asin(r64963);
        double r64965 = sin(r64964);
        double r64966 = r64965 * r64959;
        double r64967 = r64958 - r64966;
        double r64968 = exp(r64967);
        double r64969 = log(r64968);
        double r64970 = atan2(r64957, r64969);
        double r64971 = r64949 + r64970;
        return r64971;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1}}\]
  3. Using strategy rm

  4. Applied add-log-exp0.2

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

  5. Applied add-log-exp0.2

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

  6. Applied diff-log0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))