Average Error: 1.0 → 1.0
Time: 22.6s
Precision: binary64
Cost: 45568
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, \cos \phi_1\right)} \]
(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 lambda2)) (cos phi2))
   (fma (cos (- lambda1 lambda2)) (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 - lambda2)) * cos(phi2)), fma(cos((lambda1 - lambda2)), 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(sin(Float64(lambda1 - lambda2)) * cos(phi2)), fma(cos(Float64(lambda1 - lambda2)), 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[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $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{\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)}

Error

Derivation

  1. Initial program 1.0

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

  2. Simplified1.0

    \[\leadsto \color{blue}{\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}} \]
    Proof
    (+.f64 lambda1 (atan2.f64 (*.f64 (sin.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2)) (fma.f64 (cos.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2) (cos.f64 phi1)))): 0 points increase in error, 0 points decrease in error
    (+.f64 lambda1 (atan2.f64 (Rewrite<= *-commutative (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2)))) (fma.f64 (cos.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2) (cos.f64 phi1)))): 0 points increase in error, 0 points decrease in error
    (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (Rewrite<= fma-def (+.f64 (*.f64 (cos.f64 (-.f64 lambda1 lambda2)) (cos.f64 phi2)) (cos.f64 phi1))))): 3 points increase in error, 1 points decrease in error
    (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (+.f64 (Rewrite<= *-commutative (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2)))) (cos.f64 phi1)))): 0 points increase in error, 0 points decrease in error
    (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (Rewrite<= +-commutative (+.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2))))))): 0 points increase in error, 0 points decrease in error

  3. Final simplification1.0

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

Alternatives

Alternative 1
Error7.9
Cost45960
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_1 \leq -0.62:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.9948:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{t_0 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot t_0 + 1}\\ \end{array} \]
Alternative 2
Error8.1
Cost45832
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_1 \leq -0.62:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.9948:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]
Alternative 3
Error8.1
Cost45832
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_1 \leq -0.62:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.9948:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]
Alternative 4
Error6.7
Cost39812
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_1 \leq 0.9948:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 + t_0 \cdot \left(1 + \phi_2 \cdot \left(\phi_2 \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot t_0 + 1}\\ \end{array} \]
Alternative 5
Error1.5
Cost39368
\[\begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{if}\;\lambda_2 \leq -1.0761621076095574 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 8.334311823640215 \cdot 10^{-39}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error1.0
Cost39296
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 7
Error7.7
Cost39172
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\cos \phi_2 \leq 0.9998:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_2}\\ \end{array} \]
Alternative 8
Error1.4
Cost39168
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2} \]
Alternative 9
Error11.0
Cost32904
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_2 + 1}\\ \mathbf{if}\;\lambda_2 \leq -0.000771194973683657:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.9335887413924528:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error17.6
Cost26632
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + 1}\\ \mathbf{if}\;\phi_2 \leq -4.836366055100444 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 6.016124785748582 \cdot 10^{+22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error17.6
Cost26504
\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + 1}\\ \mathbf{if}\;\phi_2 \leq -36043.41177576715:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 6.016124785748582 \cdot 10^{+22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \lambda_2 + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 12
Error20.6
Cost26240
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_2 + 1} \]
Alternative 13
Error29.0
Cost26112
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{1 + \cos \lambda_1} \]
Alternative 14
Error29.0
Cost19712
\[\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{1 + \cos \left(\lambda_2 - \lambda_1\right)} \]

Error

Reproduce

herbie shell --seed 2022296 
(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)))))))