Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.0% → 99.9%

Time: 5.4s

Alternatives: 10

Speedup: 1.3×

• 30.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 59.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

4. Applied rewrites 95.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-add N/A

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

   5. cos-sum N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in phi1 around inf

   \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 99.9

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

9. Applied rewrites 99.9%

   \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right) - \color{blue}{\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

10. Taylor expanded in phi1 around inf

    \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)} - \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lift-*.f64 99.9

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

12. Applied rewrites 99.9%

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

Alternative 2: 95.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

4. Applied rewrites 95.9%

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

Alternative 3: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0175:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1 - \phi_2, \cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.0175)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi1)) (- lambda1 lambda2))) R)
   (* (hypot (- phi1 phi2) (* (cos (* 0.5 phi2)) (- lambda1 lambda2))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi1 - phi2), (Math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.0175:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi1)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi1 - phi2), (math.cos((0.5 * phi2)) * (lambda1 - lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.0175)
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi1 - phi2), Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.0175)
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi1 - phi2), (cos((0.5 * phi2)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.0175], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.017500000000000002

   1. Initial program 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   if 0.017500000000000002 < phi2

   1. Initial program 52.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

Alternative 4: 90.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

4. Applied rewrites 95.9%

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

5. Taylor expanded in phi1 around inf

   \[\leadsto \mathsf{hypot}\left(\phi_1 - \phi_2, \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
6. Step-by-step derivation

   1. lower-*.f64 90.4

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

7. Applied rewrites 90.4%

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

Alternative 5: 33.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8 \cdot 10^{+196}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 8e+196)
   (* R (fma -1.0 phi1 phi2))
   (* (* R lambda2) (cos (* 0.5 (+ phi2 phi1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 8e+196) {
		tmp = R * fma(-1.0, phi1, phi2);
	} else {
		tmp = (R * lambda2) * cos((0.5 * (phi2 + phi1)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 8e+196)
		tmp = Float64(R * fma(-1.0, phi1, phi2));
	else
		tmp = Float64(Float64(R * lambda2) * cos(Float64(0.5 * Float64(phi2 + phi1))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 8e+196], N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision], N[(N[(R * lambda2), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 7.9999999999999996e196

   1. Initial program 60.4%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

      6. lower-/.f64 28.6

         \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

   5. Applied rewrites 28.6%

      \[\leadsto R \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right)} \]

   6. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto R \cdot \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(\left(\mathsf{neg}\left(\phi_1\right)\right) + \phi_2\right) \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]

      4. lower-fma.f64 31.6

         \[\leadsto R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \]

   8. Applied rewrites 31.6%

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

   if 7.9999999999999996e196 < lambda2

   1. Initial program 45.5%

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

   3. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right) \]

      7. lower-+.f64 48.1

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 30.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -1e+135)
   (* (fma (/ (* phi1 R) phi2) -1.0 R) phi2)
   (* R (fma -1.0 phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1e+135) {
		tmp = fma(((phi1 * R) / phi2), -1.0, R) * phi2;
	} else {
		tmp = R * fma(-1.0, phi1, phi2);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e+135)
		tmp = Float64(fma(Float64(Float64(phi1 * R) / phi2), -1.0, R) * phi2);
	else
		tmp = Float64(R * fma(-1.0, phi1, phi2));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e+135], N[(N[(N[(N[(phi1 * R), $MachinePrecision] / phi2), $MachinePrecision] * -1.0 + R), $MachinePrecision] * phi2), $MachinePrecision], N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -9.99999999999999962e134

   1. Initial program 45.3%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{R \cdot \phi_1}{\phi_2}, -1, R\right) \cdot \phi_2 \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{R \cdot \phi_1}{\phi_2}, -1, R\right) \cdot \phi_2 \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2 \]

      8. lower-*.f64 21.0

         \[\leadsto \mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2 \]

   5. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\phi_1 \cdot R}{\phi_2}, -1, R\right) \cdot \phi_2} \]

   if -9.99999999999999962e134 < (-.f64 lambda1 lambda2)

   1. Initial program 63.5%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

      6. lower-/.f64 30.3

         \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

   5. Applied rewrites 30.3%

      \[\leadsto R \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right)} \]

   6. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto R \cdot \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(\left(\mathsf{neg}\left(\phi_1\right)\right) + \phi_2\right) \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]

      4. lower-fma.f64 33.6

         \[\leadsto R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \]

   8. Applied rewrites 33.6%

      \[\leadsto R \cdot \mathsf{fma}\left(-1, \color{blue}{\phi_1}, \phi_2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 30.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -8.5 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(R \cdot \frac{\phi_1}{\phi_2}, -1, R\right) \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -8.5e+134)
   (* (fma (* R (/ phi1 phi2)) -1.0 R) phi2)
   (* R (fma -1.0 phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -8.5e+134) {
		tmp = fma((R * (phi1 / phi2)), -1.0, R) * phi2;
	} else {
		tmp = R * fma(-1.0, phi1, phi2);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -8.5e+134)
		tmp = Float64(fma(Float64(R * Float64(phi1 / phi2)), -1.0, R) * phi2);
	else
		tmp = Float64(R * fma(-1.0, phi1, phi2));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -8.5e+134], N[(N[(N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision] * -1.0 + R), $MachinePrecision] * phi2), $MachinePrecision], N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -8.50000000000000024e134

   1. Initial program 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in phi2 around inf

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

   7. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(R \cdot \frac{\phi_1}{\phi_2}, -1, R\right) \cdot \phi_2} \]

   if -8.50000000000000024e134 < (-.f64 lambda1 lambda2)

   1. Initial program 63.5%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

      6. lower-/.f64 30.3

         \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

   5. Applied rewrites 30.3%

      \[\leadsto R \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right)} \]

   6. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto R \cdot \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(\left(\mathsf{neg}\left(\phi_1\right)\right) + \phi_2\right) \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]

      4. lower-fma.f64 33.6

         \[\leadsto R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \]

   8. Applied rewrites 33.6%

      \[\leadsto R \cdot \mathsf{fma}\left(-1, \color{blue}{\phi_1}, \phi_2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 29.1% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.4 \cdot 10^{-51}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.4e-51) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.4e-51) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 <= (-6.4d-51)) then
        tmp = r * -phi1
    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 <= -6.4e-51) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.4e-51:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.4e-51)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6.4e-51)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.4e-51], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.4e-51

   1. Initial program 54.5%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

         \[\leadsto R \cdot \left(\mathsf{neg}\left(\phi_1\right)\right) \]

      2. lower-neg.f64 53.6

         \[\leadsto R \cdot \left(-\phi_1\right) \]

   5. Applied rewrites 53.6%

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

   if -6.4e-51 < phi1

   1. Initial program 60.9%

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

   3. Taylor expanded in phi2 around inf

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

   5. Applied rewrites 18.8%

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

Alternative 9: 30.3% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (fma -1.0 phi1 phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * fma(-1.0, phi1, phi2);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * fma(-1.0, phi1, phi2))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(-1.0 * phi1 + phi2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in phi2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

   6. lower-/.f64 27.5

      \[\leadsto R \cdot \left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right) \]

5. Applied rewrites 27.5%

   \[\leadsto R \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\phi_1}{\phi_2}, -1, 1\right) \cdot \phi_2\right)} \]

6. Taylor expanded in phi1 around 0

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

   1. mul-1-neg N/A

      \[\leadsto R \cdot \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto R \cdot \left(\left(\mathsf{neg}\left(\phi_1\right)\right) + \phi_2\right) \]

   3. mul-1-neg N/A

      \[\leadsto R \cdot \left(-1 \cdot \phi_1 + \phi_2\right) \]

   4. lower-fma.f64 30.3

      \[\leadsto R \cdot \mathsf{fma}\left(-1, \phi_1, \phi_2\right) \]

8. Applied rewrites 30.3%

   \[\leadsto R \cdot \mathsf{fma}\left(-1, \color{blue}{\phi_1}, \phi_2\right) \]
9. Add Preprocessing

Alternative 10: 17.5% accurate, 46.5× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 59.0%

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

3. Taylor expanded in phi2 around inf

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

5. Applied rewrites 17.5%

   \[\leadsto R \cdot \color{blue}{\phi_2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025088 
(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))))))