?

Average Error: 98.6% → 99.6%
Time: 33.6s
Precision: binary64
Cost: 84480.00

?

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1, \cos \phi_2, \cos \phi_1\right)} \]
(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
 (+
  lambda1
  (atan2
   (*
    (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
    (cos phi2))
   (fma
    (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos lambda1)))
    (cos phi2)
    (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) {
	return lambda1 + atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), fma(((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))), cos(phi2), cos(phi1)));
}

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

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_] := N[(lambda1 + N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 98.6

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

  2. Simplified98.7

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

    [Start]98.6

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

    *-commutative [=>]98.6

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

    +-commutative [=>]98.6

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

    *-commutative [=>]98.6

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

    fma-def [=>]98.7

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

  3. Applied egg-rr98.7

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

  4. Applied egg-rr99.6

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

  5. Final simplification99.6

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

Alternatives

Alternative 1
Error98.9%
Cost78080.00
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \lambda_1 + \lambda_1 \cdot \sin \lambda_2, \cos \phi_2, \cos \phi_1\right)} \]
Alternative 2
Error98.0%
Cost65280.00
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)} \]
Alternative 3
Error98.7%
Cost65024.00
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, \cos \phi_1\right)} \]
Alternative 4
Error98.0%
Cost52352.00
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)} \]
Alternative 5
Error98.7%
Cost45568.00
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, \cos \phi_1\right)} \]
Alternative 6
Error98.6%
Cost39296.00
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 7
Error88.3%
Cost39172.00
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.9999998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
Alternative 8
Error97.8%
Cost39168.00
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_2 \cdot \cos \phi_2} \]
Alternative 9
Error88.4%
Cost33544.00
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0008:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 3.6:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]
Alternative 10
Error80.3%
Cost33284.00
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.44:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{\left(t_0 + 1\right) + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 + t_0}\\ \end{array} \]
Alternative 11
Error88.3%
Cost33160.00
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0004:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 3.6:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot t_0 + 1}\\ \end{array} \]
Alternative 12
Error88.2%
Cost33033.00
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0003 \lor \neg \left(\phi_2 \leq 3.6\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
Alternative 13
Error78.0%
Cost32900.00
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.44:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
Alternative 14
Error70.8%
Cost26376.00
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -2.5 \cdot 10^{+51}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\left(\cos \left(\lambda_2 - \lambda_1\right) + 2\right) + -1}\\ \mathbf{elif}\;\lambda_2 \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\left(\cos \left(\lambda_1 - \lambda_2\right) + 1\right) + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)}\\ \end{array} \]
Alternative 15
Error71.4%
Cost26372.00
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.984:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
Alternative 16
Error71.2%
Cost26244.00
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.984:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_2 + 1}\\ \end{array} \]
Alternative 17
Error76.5%
Cost26240.00
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 18
Error76.1%
Cost26112.00
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1} \]
Alternative 19
Error69.8%
Cost20488.00
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -5.4 \cdot 10^{+51}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\left(\cos \left(\lambda_2 - \lambda_1\right) + 2\right) + -1}\\ \mathbf{elif}\;\lambda_2 \leq 1.36 \cdot 10^{-120}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{1 + \left(\cos \phi_1 + \lambda_1 \cdot \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\left(\cos \left(\lambda_1 - \lambda_2\right) + 1\right) + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)}\\ \end{array} \]
Alternative 20
Error57.0%
Cost19977.00
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.6 \cdot 10^{-112} \lor \neg \left(\lambda_2 \leq 10^{-56}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\lambda_1 \cdot \sin \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
Alternative 21
Error54.5%
Cost19712.00
\[\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{1 + \cos \left(\lambda_2 - \lambda_1\right)} \]
Alternative 22
Error66.3%
Cost19712.00
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + 1} \]

Error

Reproduce?

herbie shell --seed 2023093 
(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)))))))