Average Error: 0.9 → 0.9
Time: 23.9s
Precision: binary64
Cost: 45568
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)} \]
(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
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (fma (cos phi2) (cos (- lambda1 lambda2)) (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((cos(phi2) * sin((lambda1 - lambda2))), fma(cos(phi2), cos((lambda1 - lambda2)), 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(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(cos(phi2), cos(Float64(lambda1 - lambda2)), 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[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $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{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}

Error

Derivation

  1. Initial program 0.9

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

  2. Simplified0.9

    \[\leadsto \color{blue}{\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}} \]
    Proof
    (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (fma.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<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 phi2) (cos.f64 (-.f64 lambda1 lambda2))) (cos.f64 phi1))))): 1 points increase in error, 0 points decrease in error
    (+.f64 lambda1 (atan2.f64 (*.f64 (cos.f64 phi2) (sin.f64 (-.f64 lambda1 lambda2))) (Rewrite<= +-commutative_binary64 (+.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 simplification0.9

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

Alternatives

Alternative 1
Error1.5
Cost39432
\[\begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{if}\;\lambda_2 \leq -40270945077299175000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 3.012504338835129 \cdot 10^{-49}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error8.0
Cost39428
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.96:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{t_0 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{1 + \cos \phi_2 \cdot t_0}\\ \end{array} \]
Alternative 3
Error5.1
Cost39368
\[\begin{array}{l} t_0 := \cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{t_0}\\ \mathbf{if}\;\lambda_2 \leq -2.6199881646206165 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 6.583762951956383 \cdot 10^{-164}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error8.3
Cost39300
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.96:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]
Alternative 5
Error0.9
Cost39296
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
Alternative 6
Error17.6
Cost39172
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.91:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
Alternative 7
Error1.3
Cost39168
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2} \]
Alternative 8
Error7.2
Cost33412
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -24234.632695243763:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 86.21014337318651:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_2}\\ \end{array} \]
Alternative 9
Error7.7
Cost33032
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \mathbf{if}\;\phi_2 \leq -0.002614100320759722:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 5.363865857220193 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error14.0
Cost32640
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_2} \]
Alternative 11
Error20.6
Cost26368
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_2 - \lambda_1\right)} \]
Alternative 12
Error20.7
Cost26240
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + 1} \]
Alternative 13
Error21.5
Cost19840
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_2 - \lambda_1\right)} \]
Alternative 14
Error21.6
Cost19712
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + 1} \]

Error

Reproduce

herbie shell --seed 2022290 
(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)))))))