Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.8% → 99.9%

Time: 21.4s

Alternatives: 15

Speedup: 3.0×

• 30.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right), \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \left(-\sin \left(0.5 \cdot \phi_1\right)\right)\right)\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (fma
     (cos (* phi2 0.5))
     (cos (* 0.5 phi1))
     (log1p (expm1 (* (sin (* phi2 0.5)) (- (sin (* 0.5 phi1))))))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * fma(cos((phi2 * 0.5)), cos((0.5 * phi1)), log1p(expm1((sin((phi2 * 0.5)) * -sin((0.5 * phi1))))))), (phi1 - phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * fma(cos(Float64(phi2 * 0.5)), cos(Float64(0.5 * phi1)), log1p(expm1(Float64(sin(Float64(phi2 * 0.5)) * Float64(-sin(Float64(0.5 * phi1)))))))), Float64(phi1 - phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[Log[1 + N[(Exp[N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

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

   1. hypot-def 96.3%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 96.3%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Step-by-step derivation

   1. add-log-exp 96.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

   2. div-inv 96.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

   3. metadata-eval 96.2%

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

5. Applied egg-rr 96.2%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 96.2%

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

   2. +-commutative 96.2%

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

   3. distribute-lft-in 96.2%

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

   4. cos-sum 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

   8. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]
8. Step-by-step derivation

   1. add-log-exp 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. fma-def 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

    1. log1p-expm1-u 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (fma
     (cos (* phi2 0.5))
     (cos (* 0.5 phi1))
     (* (sin (* phi2 0.5)) (- (sin (* 0.5 phi1))))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * fma(cos((phi2 * 0.5)), cos((0.5 * phi1)), (sin((phi2 * 0.5)) * -sin((0.5 * phi1))))), (phi1 - phi2));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. hypot-def 96.3%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 96.3%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Step-by-step derivation

   1. add-log-exp 96.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

   2. div-inv 96.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

   3. metadata-eval 96.2%

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

5. Applied egg-rr 96.2%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 96.2%

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

   2. +-commutative 96.2%

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

   3. distribute-lft-in 96.2%

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

   4. cos-sum 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

   8. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]
8. Step-by-step derivation

   1. add-log-exp 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. fma-def 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot t_0, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot t_0, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (-
          (* (cos (* phi2 0.5)) (cos (* 0.5 phi1)))
          (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))))))
   (if (<= lambda1 -4.6e+16)
     (* R (hypot (* lambda1 t_0) (- phi1 phi2)))
     (* R (hypot (* lambda2 t_0) (- phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)));
	double tmp;
	if (lambda1 <= -4.6e+16) {
		tmp = R * hypot((lambda1 * t_0), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * t_0), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.cos((phi2 * 0.5)) * Math.cos((0.5 * phi1))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1)));
	double tmp;
	if (lambda1 <= -4.6e+16) {
		tmp = R * Math.hypot((lambda1 * t_0), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * t_0), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.cos((phi2 * 0.5)) * math.cos((0.5 * phi1))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1)))
	tmp = 0
	if lambda1 <= -4.6e+16:
		tmp = R * math.hypot((lambda1 * t_0), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * t_0), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))))
	tmp = 0.0
	if (lambda1 <= -4.6e+16)
		tmp = Float64(R * hypot(Float64(lambda1 * t_0), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * t_0), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)));
	tmp = 0.0;
	if (lambda1 <= -4.6e+16)
		tmp = R * hypot((lambda1 * t_0), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * t_0), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -4.6e+16], N[(R * N[Sqrt[N[(lambda1 * t$95$0), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * t$95$0), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.6e16

   1. Initial program 54.6%

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

      1. hypot-def 95.2%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 95.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

      2. div-inv 95.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

      3. metadata-eval 95.1%

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

   5. Applied egg-rr 95.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. distribute-lft-in 95.1%

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

      4. cos-sum 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. *-commutative 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in lambda1 around inf 94.3%

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

   if -4.6e16 < lambda1

   1. Initial program 70.2%

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

      1. hypot-def 96.6%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 96.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

      2. div-inv 96.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

      3. metadata-eval 96.6%

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

   5. Applied egg-rr 96.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 96.6%

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

      2. +-commutative 96.6%

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

      3. distribute-lft-in 96.6%

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

      4. cos-sum 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. add-log-exp 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. fma-def 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in lambda1 around 0 63.1%

       \[\leadsto \color{blue}{\sqrt{{\left(\phi_1 - \phi_2\right)}^{2} + {\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} \cdot {\lambda_2}^{2}} \cdot R} \]
   11. Step-by-step derivation

       1. *-commutative 63.1%

          \[\leadsto \color{blue}{R \cdot \sqrt{{\left(\phi_1 - \phi_2\right)}^{2} + {\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} \cdot {\lambda_2}^{2}}} \]

       2. +-commutative 63.1%

          \[\leadsto R \cdot \sqrt{\color{blue}{{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} \cdot {\lambda_2}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]

       3. unpow2 63.1%

          \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right)} \cdot {\lambda_2}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}} \]

       4. unpow2 63.1%

          \[\leadsto R \cdot \sqrt{\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right) \cdot \color{blue}{\left(\lambda_2 \cdot \lambda_2\right)} + {\left(\phi_1 - \phi_2\right)}^{2}} \]

       5. unswap-sqr 63.1%

          \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \lambda_2\right) \cdot \left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \lambda_2\right)} + {\left(\phi_1 - \phi_2\right)}^{2}} \]

       6. unpow2 63.1%

          \[\leadsto R \cdot \sqrt{\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \lambda_2\right) \cdot \left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \lambda_2\right) + \color{blue}{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}} \]

   12. Simplified 88.1%

       \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (-
     (* (cos (* phi2 0.5)) (cos (* 0.5 phi1)))
     (* (sin (* phi2 0.5)) (sin (* 0.5 phi1)))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * ((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1))))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * ((Math.cos((phi2 * 0.5)) * Math.cos((0.5 * phi1))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1))))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * ((math.cos((phi2 * 0.5)) * math.cos((0.5 * phi1))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1))))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * ((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1))))), (phi1 - phi2));
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. hypot-def 96.3%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 96.3%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Step-by-step derivation

   1. add-log-exp 96.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

   2. div-inv 96.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

   3. metadata-eval 96.2%

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

5. Applied egg-rr 96.2%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 96.2%

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

   2. +-commutative 96.2%

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

   3. distribute-lft-in 96.2%

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

   4. cos-sum 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

   8. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in phi2 around inf 99.9%

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

9. Final simplification 99.9%

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

Alternative 5: 82.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.8e-33)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.8e-33) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.8e-33) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.8e-33:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.8e-33)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.8e-33)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.8e-33], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.8e-33

   1. Initial program 69.0%

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

      1. hypot-def 93.3%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around 0 92.4%

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

   if -2.8e-33 < phi1

   1. Initial program 65.6%

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

      1. hypot-def 97.5%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 60.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \]

      2. *-commutative 60.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)\right), \phi_1 - \phi_2\right) \]

      3. div-inv 60.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \phi_1 - \phi_2\right) \]

      4. metadata-eval 60.8%

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

   5. Applied egg-rr 60.8%

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

   6. Taylor expanded in phi1 around 0 55.4%

      \[\leadsto \color{blue}{\sqrt{{\phi_2}^{2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \cdot R} \]
   7. Step-by-step derivation

      1. *-commutative 55.4%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      2. unpow2 55.4%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. *-commutative 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + {\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)}}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      5. unpow2 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      6. unswap-sqr 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      7. *-commutative 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \]

      8. *-commutative 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \]

      9. hypot-def 80.9%

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

      10. *-commutative 80.9%

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

   8. Simplified 80.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5.8e-9)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.8e-9) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.8e-9) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5.8e-9:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.8e-9)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5.8e-9)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5.8e-9], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.79999999999999982e-9

   1. Initial program 68.0%

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

      1. hypot-def 97.2%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around 0 95.7%

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

   if 5.79999999999999982e-9 < phi2

   1. Initial program 62.0%

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

      1. hypot-def 92.9%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 92.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 92.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 7: 95.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. hypot-def 96.3%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 96.3%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

4. Final simplification 96.3%

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

Alternative 8: 75.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.2e+50)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.2e+50) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.2e+50) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.2e+50:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.2e+50)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.2e+50)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.2e+50], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.1999999999999999e50

   1. Initial program 68.8%

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

      1. hypot-def 97.0%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 60.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \]

      2. *-commutative 60.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)\right), \phi_1 - \phi_2\right) \]

      3. div-inv 60.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \phi_1 - \phi_2\right) \]

      4. metadata-eval 60.9%

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

   5. Applied egg-rr 60.9%

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

   6. Taylor expanded in phi2 around 0 59.1%

      \[\leadsto \color{blue}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \cdot R} \]
   7. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]

      2. +-commutative 59.1%

         \[\leadsto R \cdot \sqrt{\color{blue}{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      3. unpow2 59.1%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 59.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      5. unpow2 59.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      6. unswap-sqr 59.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      7. hypot-def 78.9%

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

   8. Simplified 78.9%

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

   if 4.1999999999999999e50 < phi2

   1. Initial program 57.6%

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

      1. hypot-def 93.3%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 60.9%

      \[\leadsto \color{blue}{R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. *-commutative 60.9%

         \[\leadsto \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]

      2. associate-*r* 60.9%

         \[\leadsto \phi_2 \cdot R + \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      3. distribute-rgt-out 65.0%

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)} \]

      4. mul-1-neg 65.0%

         \[\leadsto R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

      5. unsub-neg 65.0%

         \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]

   6. Simplified 65.0%

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.2 \cdot 10^{+50}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 9: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.1e-29)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.1e-29) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.1e-29) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.1e-29:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.1e-29)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.1e-29)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.1e-29], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.09999999999999989e-29

   1. Initial program 69.0%

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

      1. hypot-def 93.3%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 57.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \]

      2. *-commutative 57.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)\right), \phi_1 - \phi_2\right) \]

      3. div-inv 57.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \phi_1 - \phi_2\right) \]

      4. metadata-eval 57.7%

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

   5. Applied egg-rr 57.7%

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

   6. Taylor expanded in phi2 around 0 66.8%

      \[\leadsto \color{blue}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}} \cdot R} \]
   7. Step-by-step derivation

      1. *-commutative 66.8%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]

      2. +-commutative 66.8%

         \[\leadsto R \cdot \sqrt{\color{blue}{{\phi_1}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      3. unpow2 66.8%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 66.8%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      5. unpow2 66.8%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      6. unswap-sqr 66.8%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      7. hypot-def 88.1%

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

   8. Simplified 88.1%

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

   if -2.09999999999999989e-29 < phi1

   1. Initial program 65.6%

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

      1. hypot-def 97.5%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 60.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \]

      2. *-commutative 60.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)\right), \phi_1 - \phi_2\right) \]

      3. div-inv 60.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \phi_1 - \phi_2\right) \]

      4. metadata-eval 60.8%

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

   5. Applied egg-rr 60.8%

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

   6. Taylor expanded in phi1 around 0 55.4%

      \[\leadsto \color{blue}{\sqrt{{\phi_2}^{2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \cdot R} \]
   7. Step-by-step derivation

      1. *-commutative 55.4%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      2. unpow2 55.4%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\cos \left(0.5 \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. *-commutative 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + {\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)}}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      4. unpow2 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      5. unpow2 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      6. unswap-sqr 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      7. *-commutative 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)} \]

      8. *-commutative 55.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \]

      9. hypot-def 80.9%

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

      10. *-commutative 80.9%

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

   8. Simplified 80.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 70.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{+50}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.3e+50)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.3e+50) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.3e+50) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.3e+50:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.3e+50)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.3e+50)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.3e+50], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.3e50

   1. Initial program 68.8%

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

      1. hypot-def 97.0%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 90.7%

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

   5. Taylor expanded in phi2 around 0 57.2%

      \[\leadsto \color{blue}{\sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \cdot R} \]
   6. Step-by-step derivation

      1. *-commutative 57.2%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      2. unpow2 57.2%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. unpow2 57.2%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      4. hypot-def 72.7%

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

   7. Simplified 72.7%

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

   if 3.3e50 < phi2

   1. Initial program 57.6%

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

      1. hypot-def 93.3%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 60.9%

      \[\leadsto \color{blue}{R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. *-commutative 60.9%

         \[\leadsto \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]

      2. associate-*r* 60.9%

         \[\leadsto \phi_2 \cdot R + \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      3. distribute-rgt-out 65.0%

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)} \]

      4. mul-1-neg 65.0%

         \[\leadsto R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

      5. unsub-neg 65.0%

         \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]

   6. Simplified 65.0%

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{+50}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 11: 32.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-261}:\\ \;\;\;\;\left(\phi_1 \cdot -0.25\right) \cdot \left(\left(R \cdot \phi_2\right) \cdot \left(\lambda_2 - \lambda_1\right)\right) + \left(-0.125 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.2e-15)
   (* R (- phi2 phi1))
   (if (<= phi1 4e-261)
     (+
      (* (* phi1 -0.25) (* (* R phi2) (- lambda2 lambda1)))
      (* (+ (* -0.125 (* phi1 phi1)) 1.0) (* R (- lambda2 lambda1))))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.2e-15) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 4e-261) {
		tmp = ((phi1 * -0.25) * ((R * phi2) * (lambda2 - lambda1))) + (((-0.125 * (phi1 * phi1)) + 1.0) * (R * (lambda2 - lambda1)));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.2d-15)) then
        tmp = r * (phi2 - phi1)
    else if (phi1 <= 4d-261) then
        tmp = ((phi1 * (-0.25d0)) * ((r * phi2) * (lambda2 - lambda1))) + ((((-0.125d0) * (phi1 * phi1)) + 1.0d0) * (r * (lambda2 - lambda1)))
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.2e-15) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 4e-261) {
		tmp = ((phi1 * -0.25) * ((R * phi2) * (lambda2 - lambda1))) + (((-0.125 * (phi1 * phi1)) + 1.0) * (R * (lambda2 - lambda1)));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.2e-15:
		tmp = R * (phi2 - phi1)
	elif phi1 <= 4e-261:
		tmp = ((phi1 * -0.25) * ((R * phi2) * (lambda2 - lambda1))) + (((-0.125 * (phi1 * phi1)) + 1.0) * (R * (lambda2 - lambda1)))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.2e-15)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= 4e-261)
		tmp = Float64(Float64(Float64(phi1 * -0.25) * Float64(Float64(R * phi2) * Float64(lambda2 - lambda1))) + Float64(Float64(Float64(-0.125 * Float64(phi1 * phi1)) + 1.0) * Float64(R * Float64(lambda2 - lambda1))));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.2e-15)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= 4e-261)
		tmp = ((phi1 * -0.25) * ((R * phi2) * (lambda2 - lambda1))) + (((-0.125 * (phi1 * phi1)) + 1.0) * (R * (lambda2 - lambda1)));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.2e-15], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e-261], N[(N[(N[(phi1 * -0.25), $MachinePrecision] * N[(N[(R * phi2), $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.125 * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.19999999999999986e-15

   1. Initial program 68.2%

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

      1. hypot-def 93.2%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 65.8%

      \[\leadsto \color{blue}{R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]

      2. associate-*r* 65.8%

         \[\leadsto \phi_2 \cdot R + \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      3. distribute-rgt-out 68.4%

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)} \]

      4. mul-1-neg 68.4%

         \[\leadsto R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

      5. unsub-neg 68.4%

         \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]

   6. Simplified 68.4%

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]

   if -2.19999999999999986e-15 < phi1 < 3.99999999999999994e-261

   1. Initial program 66.0%

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

      1. hypot-def 99.9%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in lambda1 around -inf 38.9%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right) + \lambda_2 \cdot \left(R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 38.9%

         \[\leadsto \color{blue}{\lambda_2 \cdot \left(R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right) + -1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right)} \]

      2. mul-1-neg 38.9%

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

      3. unsub-neg 38.9%

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

      4. *-commutative 38.9%

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

      5. *-commutative 38.9%

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

      6. associate-*l* 38.9%

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

      7. *-commutative 38.9%

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

      8. +-commutative 38.9%

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

      9. *-commutative 38.9%

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

      10. associate-*l* 38.9%

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

      11. +-commutative 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in phi2 around 0 25.9%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot R\right) + \left(-0.5 \cdot \left(\lambda_2 \cdot \left(R \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) - -0.5 \cdot \left(R \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right)\right) \cdot \phi_2\right) - R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 25.9%

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

      2. +-commutative 25.9%

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

      3. *-commutative 25.9%

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

      4. *-commutative 25.9%

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

      5. associate-*r* 25.9%

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

      6. associate--l+ 25.9%

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

   9. Simplified 26.1%

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

   10. Taylor expanded in phi1 around 0 27.2%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 - \lambda_1\right) + \left(-0.25 \cdot \left(\phi_1 \cdot \left(R \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \phi_2\right)\right)\right) + -0.125 \cdot \left({\phi_1}^{2} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative 27.2%

          \[\leadsto \color{blue}{\left(-0.25 \cdot \left(\phi_1 \cdot \left(R \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \phi_2\right)\right)\right) + -0.125 \cdot \left({\phi_1}^{2} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)\right) + R \cdot \left(\lambda_2 - \lambda_1\right)} \]

       2. *-commutative 27.2%

          \[\leadsto \left(-0.25 \cdot \left(\phi_1 \cdot \left(R \cdot \color{blue}{\left(\phi_2 \cdot \left(\lambda_2 - \lambda_1\right)\right)}\right)\right) + -0.125 \cdot \left({\phi_1}^{2} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)\right) + R \cdot \left(\lambda_2 - \lambda_1\right) \]

       3. associate-+l+ 27.2%

          \[\leadsto \color{blue}{-0.25 \cdot \left(\phi_1 \cdot \left(R \cdot \left(\phi_2 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)\right) + \left(-0.125 \cdot \left({\phi_1}^{2} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + R \cdot \left(\lambda_2 - \lambda_1\right)\right)} \]

       4. associate-*r* 27.2%

          \[\leadsto \color{blue}{\left(-0.25 \cdot \phi_1\right) \cdot \left(R \cdot \left(\phi_2 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)} + \left(-0.125 \cdot \left({\phi_1}^{2} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + R \cdot \left(\lambda_2 - \lambda_1\right)\right) \]

       5. associate-*r* 28.6%

          \[\leadsto \left(-0.25 \cdot \phi_1\right) \cdot \color{blue}{\left(\left(R \cdot \phi_2\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)} + \left(-0.125 \cdot \left({\phi_1}^{2} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + R \cdot \left(\lambda_2 - \lambda_1\right)\right) \]

       6. *-commutative 28.6%

          \[\leadsto \left(-0.25 \cdot \phi_1\right) \cdot \left(\color{blue}{\left(\phi_2 \cdot R\right)} \cdot \left(\lambda_2 - \lambda_1\right)\right) + \left(-0.125 \cdot \left({\phi_1}^{2} \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + R \cdot \left(\lambda_2 - \lambda_1\right)\right) \]

       7. associate-*r* 28.6%

          \[\leadsto \left(-0.25 \cdot \phi_1\right) \cdot \left(\left(\phi_2 \cdot R\right) \cdot \left(\lambda_2 - \lambda_1\right)\right) + \left(\color{blue}{\left(-0.125 \cdot {\phi_1}^{2}\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)} + R \cdot \left(\lambda_2 - \lambda_1\right)\right) \]

       8. distribute-lft1-in 31.6%

          \[\leadsto \left(-0.25 \cdot \phi_1\right) \cdot \left(\left(\phi_2 \cdot R\right) \cdot \left(\lambda_2 - \lambda_1\right)\right) + \color{blue}{\left(-0.125 \cdot {\phi_1}^{2} + 1\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)} \]

       9. unpow2 31.6%

          \[\leadsto \left(-0.25 \cdot \phi_1\right) \cdot \left(\left(\phi_2 \cdot R\right) \cdot \left(\lambda_2 - \lambda_1\right)\right) + \left(-0.125 \cdot \color{blue}{\left(\phi_1 \cdot \phi_1\right)} + 1\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right) \]

   12. Simplified 31.6%

       \[\leadsto \color{blue}{\left(-0.25 \cdot \phi_1\right) \cdot \left(\left(\phi_2 \cdot R\right) \cdot \left(\lambda_2 - \lambda_1\right)\right) + \left(-0.125 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)} \]

   if 3.99999999999999994e-261 < phi1

   1. Initial program 65.9%

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

      1. hypot-def 96.2%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around inf 14.8%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 14.8%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   6. Simplified 14.8%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-261}:\\ \;\;\;\;\left(\phi_1 \cdot -0.25\right) \cdot \left(\left(R \cdot \phi_2\right) \cdot \left(\lambda_2 - \lambda_1\right)\right) + \left(-0.125 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12: 31.6% accurate, 36.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.6 \cdot 10^{+54}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq 2.7 \cdot 10^{-259}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.6e+54)
   (* phi1 (- R))
   (if (<= phi1 2.7e-259) (* R (- lambda2 lambda1)) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.6e+54) {
		tmp = phi1 * -R;
	} else if (phi1 <= 2.7e-259) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-5.6d+54)) then
        tmp = phi1 * -r
    else if (phi1 <= 2.7d-259) then
        tmp = r * (lambda2 - lambda1)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.6e+54) {
		tmp = phi1 * -R;
	} else if (phi1 <= 2.7e-259) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.6e+54:
		tmp = phi1 * -R
	elif phi1 <= 2.7e-259:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.6e+54)
		tmp = Float64(phi1 * Float64(-R));
	elseif (phi1 <= 2.7e-259)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5.6e+54)
		tmp = phi1 * -R;
	elseif (phi1 <= 2.7e-259)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.6e+54], N[(phi1 * (-R)), $MachinePrecision], If[LessEqual[phi1, 2.7e-259], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.6000000000000003e54

   1. Initial program 66.1%

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

      1. hypot-def 92.2%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 73.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 73.5%

         \[\leadsto \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      2. mul-1-neg 73.5%

         \[\leadsto \color{blue}{\left(-\phi_1\right)} \cdot R \]

   6. Simplified 73.5%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot R} \]

   if -5.6000000000000003e54 < phi1 < 2.69999999999999984e-259

   1. Initial program 68.0%

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

      1. hypot-def 99.5%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in lambda1 around -inf 37.6%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right) + \lambda_2 \cdot \left(R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 37.6%

         \[\leadsto \color{blue}{\lambda_2 \cdot \left(R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right) + -1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right)} \]

      2. mul-1-neg 37.6%

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

      3. unsub-neg 37.6%

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

      4. *-commutative 37.6%

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

      5. *-commutative 37.6%

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

      6. associate-*l* 37.6%

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

      7. *-commutative 37.6%

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

      8. +-commutative 37.6%

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

      9. *-commutative 37.6%

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

      10. associate-*l* 37.6%

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

      11. +-commutative 37.6%

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

   6. Simplified 37.6%

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

   7. Taylor expanded in phi2 around 0 25.8%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot R\right) + \left(-0.5 \cdot \left(\lambda_2 \cdot \left(R \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) - -0.5 \cdot \left(R \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right)\right) \cdot \phi_2\right) - R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 25.8%

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

      2. +-commutative 25.8%

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

      3. *-commutative 25.8%

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

      4. *-commutative 25.8%

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

      5. associate-*r* 25.9%

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

      6. associate--l+ 25.9%

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

   9. Simplified 26.0%

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

   10. Taylor expanded in phi1 around 0 27.6%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 - \lambda_1\right)} \]

   if 2.69999999999999984e-259 < phi1

   1. Initial program 65.9%

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

      1. hypot-def 96.2%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around inf 14.8%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 14.8%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   6. Simplified 14.8%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.6 \cdot 10^{+54}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq 2.7 \cdot 10^{-259}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 13: 35.7% accurate, 46.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.75 \cdot 10^{+105}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.75e+105) (* R (- lambda2 lambda1)) (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.75e+105) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.75d+105)) then
        tmp = r * (lambda2 - lambda1)
    else
        tmp = r * (phi2 - phi1)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.75e+105) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.75e+105:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.75e+105)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.75e+105)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.75e+105], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.74999999999999996e105

   1. Initial program 51.0%

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

      1. hypot-def 93.1%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in lambda1 around -inf 54.5%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right) + \lambda_2 \cdot \left(R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 54.5%

         \[\leadsto \color{blue}{\lambda_2 \cdot \left(R \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right) + -1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right)} \]

      2. mul-1-neg 54.5%

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

      3. unsub-neg 54.5%

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

      4. *-commutative 54.5%

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

      5. *-commutative 54.5%

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

      6. associate-*l* 54.5%

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

      7. *-commutative 54.5%

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

      8. +-commutative 54.5%

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

      9. *-commutative 54.5%

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

      10. associate-*l* 54.5%

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

      11. +-commutative 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in phi2 around 0 41.4%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot R\right) + \left(-0.5 \cdot \left(\lambda_2 \cdot \left(R \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right) - -0.5 \cdot \left(R \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right)\right) \cdot \phi_2\right) - R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 41.5%

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

      2. +-commutative 41.5%

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

      3. *-commutative 41.5%

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

      4. *-commutative 41.5%

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

      5. associate-*r* 41.5%

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

      6. associate--l+ 41.5%

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

   9. Simplified 44.6%

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

   10. Taylor expanded in phi1 around 0 58.5%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 - \lambda_1\right)} \]

   if -1.74999999999999996e105 < lambda1

   1. Initial program 69.6%

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

      1. hypot-def 96.9%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 32.3%

      \[\leadsto \color{blue}{R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]

      2. associate-*r* 32.3%

         \[\leadsto \phi_2 \cdot R + \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      3. distribute-rgt-out 33.7%

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)} \]

      4. mul-1-neg 33.7%

         \[\leadsto R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

      5. unsub-neg 33.7%

         \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]

   6. Simplified 33.7%

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.75 \cdot 10^{+105}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 14: 28.6% accurate, 54.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -54000000000000:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -54000000000000.0) (* phi1 (- R)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -54000000000000.0) {
		tmp = phi1 * -R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-54000000000000.0d0)) then
        tmp = phi1 * -r
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -54000000000000.0) {
		tmp = phi1 * -R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -54000000000000.0:
		tmp = phi1 * -R
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -54000000000000.0)
		tmp = Float64(phi1 * Float64(-R));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -54000000000000.0)
		tmp = phi1 * -R;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -54000000000000.0], N[(phi1 * (-R)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.4e13

   1. Initial program 68.1%

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

      1. hypot-def 92.6%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 70.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 70.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      2. mul-1-neg 70.8%

         \[\leadsto \color{blue}{\left(-\phi_1\right)} \cdot R \]

   6. Simplified 70.8%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot R} \]

   if -5.4e13 < phi1

   1. Initial program 66.1%

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

      1. hypot-def 97.6%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around inf 17.0%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 17.0%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   6. Simplified 17.0%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -54000000000000:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15: 17.4% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \phi_2 \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R phi2))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * phi2
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * phi2

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * phi2)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * phi2;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi2), $MachinePrecision]

Derivation

1. Initial program 66.6%

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

   1. hypot-def 96.3%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 96.3%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

4. Taylor expanded in phi2 around inf 15.0%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
5. Step-by-step derivation

   1. *-commutative 15.0%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]

6. Simplified 15.0%

   \[\leadsto \color{blue}{\phi_2 \cdot R} \]

7. Final simplification 15.0%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023229 
(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))))))