Average Error: 0.2 → 0.1
Time: 43.2s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\mathsf{fma}\left(\cos delta, \cos \left(\phi_1 + \phi_1\right), \cos delta\right) \cdot \frac{1}{2} - \left(\left(\sin \phi_1 \cdot \cos \phi_1\right) \cdot \cos theta\right) \cdot \sin delta} + \lambda_1\]
\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)}
↓
\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\mathsf{fma}\left(\cos delta, \cos \left(\phi_1 + \phi_1\right), \cos delta\right) \cdot \frac{1}{2} - \left(\left(\sin \phi_1 \cdot \cos \phi_1\right) \cdot \cos theta\right) \cdot \sin delta} + \lambda_1
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3245099 = lambda1;
        double r3245100 = theta;
        double r3245101 = sin(r3245100);
        double r3245102 = delta;
        double r3245103 = sin(r3245102);
        double r3245104 = r3245101 * r3245103;
        double r3245105 = phi1;
        double r3245106 = cos(r3245105);
        double r3245107 = r3245104 * r3245106;
        double r3245108 = cos(r3245102);
        double r3245109 = sin(r3245105);
        double r3245110 = r3245109 * r3245108;
        double r3245111 = r3245106 * r3245103;
        double r3245112 = cos(r3245100);
        double r3245113 = r3245111 * r3245112;
        double r3245114 = r3245110 + r3245113;
        double r3245115 = asin(r3245114);
        double r3245116 = sin(r3245115);
        double r3245117 = r3245109 * r3245116;
        double r3245118 = r3245108 - r3245117;
        double r3245119 = atan2(r3245107, r3245118);
        double r3245120 = r3245099 + r3245119;
        return r3245120;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r3245121 = phi1;
        double r3245122 = cos(r3245121);
        double r3245123 = theta;
        double r3245124 = sin(r3245123);
        double r3245125 = r3245122 * r3245124;
        double r3245126 = delta;
        double r3245127 = sin(r3245126);
        double r3245128 = r3245125 * r3245127;
        double r3245129 = cos(r3245126);
        double r3245130 = r3245121 + r3245121;
        double r3245131 = cos(r3245130);
        double r3245132 = fma(r3245129, r3245131, r3245129);
        double r3245133 = 0.5;
        double r3245134 = r3245132 * r3245133;
        double r3245135 = sin(r3245121);
        double r3245136 = r3245135 * r3245122;
        double r3245137 = cos(r3245123);
        double r3245138 = r3245136 * r3245137;
        double r3245139 = r3245138 * r3245127;
        double r3245140 = r3245134 - r3245139;
        double r3245141 = atan2(r3245128, r3245140);
        double r3245142 = lambda1;
        double r3245143 = r3245141 + r3245142;
        return r3245143;
}

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

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

  4. Applied add-cbrt-cube0.2

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

  5. Taylor expanded around inf 0.2

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

  6. Simplified0.2

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

  8. Applied sqr-sin0.2

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

  9. Simplified0.2

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

  10. Taylor expanded around inf 0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))