Average Error: 38.8 → 0.1
Time: 18.1s
Precision: binary64
Cost: 39872
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right) \]
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (sqrt
   (+
    (*
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
    (* (- phi1 phi2) (- phi1 phi2))))))
↓
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (fma
     (sin (* phi2 0.5))
     (- (sin (* 0.5 phi1)))
     (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))))
   (- phi1 phi2))))
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * fma(sin((phi2 * 0.5)), -sin((0.5 * phi1)), (cos((0.5 * phi1)) * cos((phi2 * 0.5))))), (phi1 - phi2));
}

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * sqrt(Float64(Float64(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))) * Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end
↓
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * fma(sin(Float64(phi2 * 0.5)), Float64(-sin(Float64(0.5 * phi1))), Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)))
end

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]) + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}
↓
R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)

Error

Derivation

  1. Initial program 38.8

    \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

  2. Simplified3.9

    \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
    Proof
    (*.f64 R (hypot.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (-.f64 phi1 phi2))): 0 points increase in error, 0 points decrease in error
    (*.f64 R (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))))): 163 points increase in error, 0 points decrease in error

  3. Applied egg-rr3.9

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right)}, \phi_1 - \phi_2\right) \]

  4. Applied egg-rr0.2

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}\right)\right), \phi_1 - \phi_2\right) \]

  5. Applied egg-rr0.1

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}, \phi_1 - \phi_2\right) \]

  6. Final simplification0.1

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right) \]

Alternatives

Alternative 1
Error3.6
Cost33540
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.5 \cdot 10^{+188}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 2
Error0.1
Cost33536
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right) \]
Alternative 3
Error6.5
Cost13700
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 4
Error3.9
Cost13696
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \]
Alternative 5
Error9.1
Cost13568
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right) \]
Alternative 6
Error45.0
Cost7372
\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq 1.35 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 4.5 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 7 \cdot 10^{+232}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error45.0
Cost7372
\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq 1.65 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 4.7 \cdot 10^{+174}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot t_0\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.8 \cdot 10^{+232}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \end{array} \]
Alternative 8
Error44.2
Cost7240
\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -4.65 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 0.0008:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error43.5
Cost7240
\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -250000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 2.25 \cdot 10^{-139}:\\ \;\;\;\;\left(R \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error48.9
Cost388
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.6 \cdot 10^{+79}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 11
Error46.8
Cost320
\[R \cdot \left(\phi_2 - \phi_1\right) \]
Alternative 12
Error54.2
Cost192
\[R \cdot \phi_2 \]

Error

Reproduce

herbie shell --seed 2022331 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))