Average Error: 38.9 → 0.1
Time: 20.9s
Precision: binary64
Cost: 39872
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\sin \left(0.5 \cdot \phi_2\right), -\sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right) \]
(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 (* 0.5 phi2))
     (- (sin (* 0.5 phi1)))
     (* (cos (* 0.5 phi1)) (cos (* 0.5 phi2)))))
   (- 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((0.5 * phi2)), -sin((0.5 * phi1)), (cos((0.5 * phi1)) * cos((0.5 * phi2))))), (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(0.5 * phi2)), Float64(-sin(Float64(0.5 * phi1))), Float64(cos(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))))), 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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * (-N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]) + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $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(0.5 \cdot \phi_2\right), -\sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)

Error

Derivation

  1. Initial program 38.9

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

  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

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

  4. Applied egg-rr0.1

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

  5. Applied egg-rr0.1

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

  6. Simplified0.1

    \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\sin \left(0.5 \cdot \phi_2\right), -\sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}, \phi_1 - \phi_2\right) \]
    Proof
    (*.f64 (-.f64 lambda1 lambda2) (fma.f64 (sin.f64 (*.f64 1/2 phi2)) (neg.f64 (sin.f64 (*.f64 1/2 phi1))) (*.f64 (cos.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (-.f64 lambda1 lambda2) (fma.f64 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi2 1/2))) (neg.f64 (sin.f64 (*.f64 1/2 phi1))) (*.f64 (cos.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (-.f64 lambda1 lambda2) (fma.f64 (sin.f64 (*.f64 phi2 1/2)) (neg.f64 (sin.f64 (*.f64 1/2 phi1))) (*.f64 (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi2 1/2))) (cos.f64 (*.f64 1/2 phi1))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (-.f64 lambda1 lambda2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (neg.f64 (sin.f64 (*.f64 1/2 phi1)))) (*.f64 (cos.f64 (*.f64 phi2 1/2)) (cos.f64 (*.f64 1/2 phi1)))))): 8 points increase in error, 8 points decrease in error
    (*.f64 (-.f64 lambda1 lambda2) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (cos.f64 (*.f64 1/2 phi1))) (*.f64 (sin.f64 (*.f64 phi2 1/2)) (neg.f64 (sin.f64 (*.f64 1/2 phi1))))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (cos.f64 (*.f64 1/2 phi1))) (-.f64 lambda1 lambda2)) (*.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (neg.f64 (sin.f64 (*.f64 1/2 phi1)))) (-.f64 lambda1 lambda2)))): 10 points increase in error, 13 points decrease in error

  7. Final simplification0.1

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

Alternatives

Alternative 1
Error0.1
Cost39872
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_2\right), -\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right) \]
Alternative 2
Error0.1
Cost33536
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right) \]
Alternative 3
Error3.9
Cost20480
\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ R \cdot \mathsf{hypot}\left(\lambda_1 \cdot t_0 - \lambda_2 \cdot t_0, \phi_1 - \phi_2\right) \end{array} \]
Alternative 4
Error3.9
Cost13696
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \phi_1 - \phi_2\right) \]
Alternative 5
Error3.9
Cost13696
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \]

Error

Reproduce

herbie shell --seed 2022334 
(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))))))