Average Error: 0.9 → 0.3
Time: 18.8s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2\\ t_1 := -\sin \lambda_2\\ t_2 := \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\\ t_3 := \cos \lambda_2 \cdot \cos \lambda_1\\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{fma}\left(t_2 \cdot t_2, t_2, \cos \lambda_1 \cdot t_1\right) + \mathsf{fma}\left(t_1, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, e^{\mathsf{log1p}\left(\frac{t_3 \cdot t_3 - t_0 \cdot t_0}{t_3 - t_0}\right)} + -1, \cos \phi_1\right)} \end{array} \]
(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 lambda1) (sin lambda2)))
        (t_1 (- (sin lambda2)))
        (t_2 (cbrt (* (cos lambda2) (sin lambda1))))
        (t_3 (* (cos lambda2) (cos lambda1))))
   (+
    lambda1
    (atan2
     (*
      (cos phi2)
      (+
       (fma (* t_2 t_2) t_2 (* (cos lambda1) t_1))
       (fma t_1 (cos lambda1) (* (cos lambda1) (sin lambda2)))))
     (fma
      (cos phi2)
      (+ (exp (log1p (/ (- (* t_3 t_3) (* t_0 t_0)) (- t_3 t_0)))) -1.0)
      (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) {
	double t_0 = sin(lambda1) * sin(lambda2);
	double t_1 = -sin(lambda2);
	double t_2 = cbrt((cos(lambda2) * sin(lambda1)));
	double t_3 = cos(lambda2) * cos(lambda1);
	return lambda1 + atan2((cos(phi2) * (fma((t_2 * t_2), t_2, (cos(lambda1) * t_1)) + fma(t_1, cos(lambda1), (cos(lambda1) * sin(lambda2))))), fma(cos(phi2), (exp(log1p((((t_3 * t_3) - (t_0 * t_0)) / (t_3 - t_0)))) + -1.0), 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)
	t_0 = Float64(sin(lambda1) * sin(lambda2))
	t_1 = Float64(-sin(lambda2))
	t_2 = cbrt(Float64(cos(lambda2) * sin(lambda1)))
	t_3 = Float64(cos(lambda2) * cos(lambda1))
	return Float64(lambda1 + atan(Float64(cos(phi2) * Float64(fma(Float64(t_2 * t_2), t_2, Float64(cos(lambda1) * t_1)) + fma(t_1, cos(lambda1), Float64(cos(lambda1) * sin(lambda2))))), fma(cos(phi2), Float64(exp(log1p(Float64(Float64(Float64(t_3 * t_3) - Float64(t_0 * t_0)) / Float64(t_3 - t_0)))) + -1.0), 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_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sin[lambda2], $MachinePrecision])}, Block[{t$95$2 = N[Power[N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$2 + N[(N[Cos[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] * N[(N[Exp[N[Log[1 + 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] + -1.0), $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)}
↓
\begin{array}{l}
t_0 := \sin \lambda_1 \cdot \sin \lambda_2\\
t_1 := -\sin \lambda_2\\
t_2 := \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\\
t_3 := \cos \lambda_2 \cdot \cos \lambda_1\\
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{fma}\left(t_2 \cdot t_2, t_2, \cos \lambda_1 \cdot t_1\right) + \mathsf{fma}\left(t_1, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, e^{\mathsf{log1p}\left(\frac{t_3 \cdot t_3 - t_0 \cdot t_0}{t_3 - t_0}\right)} + -1, \cos \phi_1\right)}
\end{array}

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

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

  3. Applied egg-rr0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \color{blue}{e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1}, \cos \phi_1\right)} \]

  4. Applied egg-rr0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, -\cos \lambda_1 \cdot \sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}}{\mathsf{fma}\left(\cos \phi_2, e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1, \cos \phi_1\right)} \]

  5. Applied egg-rr0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, -\cos \lambda_1 \cdot \sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}{\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \lambda_2}}\right)} - 1, \cos \phi_1\right)} \]

  6. Final simplification0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, e^{\mathsf{log1p}\left(\frac{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}{\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_1 \cdot \sin \lambda_2}\right)} + -1, \cos \phi_1\right)} \]

Reproduce

herbie shell --seed 2022207 
(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)))))))