Midpoint on a great circle

Average Accuracy: 98.7% → 99.6%

Time: 25.2s

Precision: binary64

Cost: 201536

Math

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

↓

\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \sin \lambda_1\\ t_1 := \cos \lambda_2 \cdot \sin \lambda_1\\ t_2 := \mathsf{fma}\left(\cos \lambda_1, \sin \lambda_2, t_1\right)\\ t_3 := \cos \lambda_1 \cdot \cos \lambda_2\\ \lambda_1 + \tan^{-1}_* \frac{\frac{t_2}{\frac{\frac{t_2}{\cos \phi_2}}{\mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t_1\right)}}}{\cos \phi_1 + \cos \phi_2 \cdot \frac{t_3 \cdot t_3 - t_0 \cdot t_0}{t_3 - t_0}} \end{array} \]

FPCore

(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 (* (sin lambda2) (sin lambda1)))
        (t_1 (* (cos lambda2) (sin lambda1)))
        (t_2 (fma (cos lambda1) (sin lambda2) t_1))
        (t_3 (* (cos lambda1) (cos lambda2))))
   (+
    lambda1
    (atan2
     (/ t_2 (/ (/ t_2 (cos phi2)) (fma (- (sin lambda2)) (cos lambda1) t_1)))
     (+
      (cos phi1)
      (* (cos phi2) (/ (- (* t_3 t_3) (* t_0 t_0)) (- t_3 t_0))))))))

C

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 = sin(lambda2) * sin(lambda1);
	double t_1 = cos(lambda2) * sin(lambda1);
	double t_2 = fma(cos(lambda1), sin(lambda2), t_1);
	double t_3 = cos(lambda1) * cos(lambda2);
	return lambda1 + atan2((t_2 / ((t_2 / cos(phi2)) / fma(-sin(lambda2), cos(lambda1), t_1))), (cos(phi1) + (cos(phi2) * (((t_3 * t_3) - (t_0 * t_0)) / (t_3 - t_0)))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

↓

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda2) * sin(lambda1))
	t_1 = Float64(cos(lambda2) * sin(lambda1))
	t_2 = fma(cos(lambda1), sin(lambda2), t_1)
	t_3 = Float64(cos(lambda1) * cos(lambda2))
	return Float64(lambda1 + atan(Float64(t_2 / Float64(Float64(t_2 / cos(phi2)) / fma(Float64(-sin(lambda2)), cos(lambda1), t_1))), Float64(cos(phi1) + Float64(cos(phi2) * Float64(Float64(Float64(t_3 * t_3) - Float64(t_0 * t_0)) / Float64(t_3 - t_0))))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(t$95$2 / N[(N[(t$95$2 / N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 98.7%

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

2. Applied egg-rr 93.8%

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

3. Simplified 98.8%

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

   [Start] 93.8	\[ \lambda_1 + \tan^{-1}_* \frac{\frac{\cos \phi_2 \cdot \left(\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right) - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \lambda_2}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   *-commutative [=>] 93.8	\[ \lambda_1 + \tan^{-1}_* \frac{\frac{\color{blue}{\left(\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right) - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}}{\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \lambda_2}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   associate-/l* [=>] 93.8	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\frac{\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right) - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_2}}}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   difference-of-squares [=>] 94.1	\[ \lambda_1 + \tan^{-1}_* \frac{\frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}}{\frac{\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_2}}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   associate-/l* [=>] 98.8	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\frac{\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \lambda_2}{\frac{\frac{\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_2}}{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}}}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	78208

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

Alternative 2
Accuracy	99.0%
Cost	71808

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

Alternative 3
Accuracy	98.8%
Cost	58752

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

Alternative 4
Accuracy	98.8%
Cost	45568

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

Alternative 5
Accuracy	88.5%
Cost	39428

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.98:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]

Alternative 6
Accuracy	88.7%
Cost	39300

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.9955:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 7
Accuracy	88.1%
Cost	39300

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.98:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_2 \cdot \cos \phi_2 + 1}\\ \end{array} \]

Alternative 8
Accuracy	98.7%
Cost	39296

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

Alternative 9
Accuracy	80.7%
Cost	39172

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.9:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) + \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_2 + \cos \phi_1}\\ \end{array} \]

Alternative 10
Accuracy	98.0%
Cost	39168

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

Alternative 11
Accuracy	89.5%
Cost	33544

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -490000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\phi_1 \leq 0.00315:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 12
Accuracy	73.4%
Cost	32905

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -490000 \lor \neg \left(\phi_1 \leq 4.8 \cdot 10^{+46}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_1 \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\cos \left(\lambda_1 - \lambda_2\right) + \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	88.1%
Cost	32905

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -6500000 \lor \neg \left(\lambda_2 \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]

Alternative 14
Accuracy	88.1%
Cost	32904

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -6500000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_2 + \cos \phi_1}\\ \end{array} \]

Alternative 15
Accuracy	70.2%
Cost	26752

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)} \]

Alternative 16
Accuracy	68.6%
Cost	26368

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

Alternative 17
Accuracy	63.4%
Cost	26240

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

Alternative 18
Accuracy	68.3%
Cost	26240

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

Alternative 19
Accuracy	55.3%
Cost	20224

\[\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{1 + \cos \left(\lambda_2 - \lambda_1\right)} \]

Alternative 20
Accuracy	55.2%
Cost	19712

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

Reproduce

herbie shell --seed 2023129 
(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)))))))